Why NPV Uses Cash Flows
Net present value measures the value, at the decision date, of the cash amounts a project is expected to generate and require. Accounting earnings are useful for reporting performance, but they are not the same as cash: they include noncash expenses such as depreciation, and they do not necessarily record a large investment outlay as an expense when the money is paid. Substituting earnings for cash flows can therefore distort both the size and timing of the amounts discounted.
The earlier material on NPV establishes that project cash flows are discounted at the required return for the relevant period. Here, the central task is to identify the correct amounts to discount. The methods below are aligned with the SOA VEE Accounting and Finance syllabus.
For an operating project, a useful starting point is cash operating contribution: revenue less cash operating costs, before depreciation and taxes. Depreciation is an accounting allocation of an asset's cost over time. It is not a payment made each year, although tax rules may allow it to reduce taxable income. Capital expenditure, by contrast, is a cash payment when the asset is acquired. These two items play different roles in an NPV calculation.
From Accounting Earnings to Operating Cash Flow
Use the following model for the operating cash flow formula. Cash operating contribution is received at each year-end; depreciation is tax-deductible; the project's taxable operating income is positive; the tax rate is constant; and taxes are paid in the same year as the taxable income. Assume the depreciation tax deduction can be used immediately, with no other tax adjustments. Financing costs are excluded from project operating cash flow. If taxable income is negative, or tax deductions cannot be used immediately, the tax calculation needs additional assumptions and this simple formula may not apply.
Let \(C_t\) be cash operating contribution in year \(t\), \(D_t\) depreciation expense for that year, and \(T\) the tax rate. Taxable operating income is \(C_t-D_t\). After-tax accounting operating income is therefore \((C_t-D_t)(1-T)\). Because depreciation is noncash, add it back to obtain operating cash flow:
The last expression shows the two components: after-tax cash operating contribution, \(C_t(1-T)\), plus the depreciation tax shield, \(TD_t\). The shield is the tax saved because deductible depreciation reduces taxable income by \(D_t\). This is not a cash receipt equal to the depreciation amount; it is a reduction in tax under the stated assumptions.
Putting Cash Flows into NPV
Assume an initial capital expenditure \(I_0\) is paid at time 0, operating cash flows occur at the ends of years 1 through \(n\), and any terminal cash flows occur at the stated time. With an effective annual required return \(i\), the NPV is the time-0 outlay plus the present values of the subsequent net project cash flows. For the simple case of one initial outlay and annual operating cash flows \(OCF_t\):
The investment is subtracted when cash is paid, not gradually as depreciation expense. If a project also requires working capital, that cash investment is included at the time it is tied up. A later recovery is a separate cash inflow when received. Depreciation can affect taxes, but it is not itself an additional cash outflow.
Worked Applications
Worked Example: Add Back Depreciation, Not the Investment
A hypothetical project requires equipment costing $100,000 at time 0. It produces annual cash operating contribution of $50,000 for four years. The equipment is depreciated straight-line to zero over those four years, taxable operating income is positive, and the tax rate is 30%. There is no salvage value or working-capital investment. The required return is 10% effective annually, and operating cash flows occur at each year-end. Find NPV. Report dollars to the cent.
Method. Calculate annual operating cash flow from after-tax operating income, add back noncash depreciation, and discount the resulting cash flows.
Annual depreciation is \(100{,}000/4=\$25{,}000\). Taxable operating income each year is \(50{,}000-25{,}000=\$25{,}000\), so tax is \(0.30(25{,}000)=\$7{,}500\). After-tax accounting operating income is \(25{,}000-7{,}500=\$17{,}500\). Adding back depreciation gives annual operating cash flow of \(17{,}500+25{,}000=\$42{,}500\).
The four discount factors are \(1/1.10=0.909091\), \(1/1.10^2=0.826446\), \(1/1.10^3=0.751315\), and \(1/1.10^4=0.683013\). Their sum is \(3.169865\). The present value of operating cash flows is \(42{,}500(3.169865446)=\$134{,}719.28\), using the unrounded factor sum for the calculation. Therefore,
Check. The project has one $100,000 cash investment at time 0 and four $42,500 year-end operating cash flows. The discounted inflows exceed the initial outflow by $34,719.28. The depreciation expense affects taxes; it is not a second investment payment.
Worked Example: Diagnose NPV Based on Accounting Earnings
A hypothetical project costs $80,000 at time 0 and generates annual cash operating contribution of $35,000 for three years. Depreciation is straight-line over the three years to zero, taxable operating income is positive, and the tax rate is 25%. There are no other cash flows. The required return is 10% effective annually, with year-end operating cash flows. Find NPV, and explain the error in discounting after-tax accounting operating income instead. Report dollars to the cent.
Method. Calculate after-tax accounting operating income, add back depreciation to obtain operating cash flow, and then discount that cash flow. Do not treat accounting earnings as cash.
Annual depreciation is \(80{,}000/3=\$26{,}666.67\) (the exact amount is \(80{,}000/3\)). Taxable operating income is \(35{,}000-80{,}000/3=\$8{,}333.33\), so annual tax is \(0.25(8{,}333.33\ldots)=\$2{,}083.33\ldots\). After-tax accounting operating income is \(8{,}333.33\ldots-2{,}083.33\ldots=\$6{,}250\). Adding back the exact depreciation amount gives annual operating cash flow \(6{,}250+80{,}000/3=\$32{,}916.67\) (rounded for display).
The three-year discount-factor sum is \(1/1.10+1/1.10^2+1/1.10^3=2.486851991\). Using unrounded amounts, the present value of the operating cash flows is \((32{,}916.6667\ldots)(2.486851991)=\$81{,}858.88\). Thus,
Common mistake. Discounting only the $6,250 of annual after-tax accounting operating income gives \(-80{,}000+6{,}250(2.486851991)=-\$64{,}457.18\). This is wrong because it omits the depreciation add-back from cash flow. The initial $80,000 outlay is still deducted at time 0; adding back depreciation does not mean adding back the equipment purchase.
Check. Each year's cash flow equals $6,250 of after-tax accounting income plus the exact $26,666.67 depreciation amount. The computed NPV is the present value of those three cash flows less the time-0 purchase price.
Worked Example: Include Working Capital and Its Recovery
A hypothetical project requires equipment costing $120,000 and working capital of $10,000 at time 0. It produces annual cash operating contribution of $60,000 for three years. Equipment depreciation is straight-line to zero over three years; taxable operating income is positive and the tax rate is 25%. The equipment has no salvage value, and the full $10,000 working-capital investment is recovered at the end of year 3. The required return is 10% effective annually. Find NPV to the cent.
Method. Find annual operating cash flow from taxable income and taxes, then include the working-capital investment and recovery at their actual cash-flow dates.
Annual depreciation is \(120{,}000/3=\$40{,}000\). Taxable operating income is \(60{,}000-40{,}000=\$20{,}000\), and tax is \(0.25(20{,}000)=\$5{,}000\). After-tax accounting operating income is \(\$15{,}000\); adding back the noncash depreciation gives annual operating cash flow of \(15{,}000+40{,}000=\$55{,}000\).
The total time-0 outlay is \(120{,}000+10{,}000=\$130{,}000\). The year-1 and year-2 operating cash flows have present value \(55{,}000(0.909090909+0.826446281)=\$95{,}454.55\). In year 3, the $55,000 operating cash flow is joined by the $10,000 recovery, making the year-3 total $65,000. Its present value is \(65{,}000(0.751314801)=\$48{,}835.46\). Therefore,
Check. The present value of all operating cash flows is \(55{,}000(2.486851991)=\$136{,}776.86\); the present value of the working-capital recovery is \(10{,}000(0.751314801)=\$7{,}513.15\). Together these are $144,290.01, which less the $130,000 time-0 outlay gives $14,290.01. The working-capital investment is not depreciated, and its recovery is not an accounting earnings item.
Common Errors and a Practical Check
The most reliable way to avoid confusing earnings with cash is to build the project cash-flow timeline before discounting. Separate amounts by their economic role: cash operating contribution, taxes, capital spending, working capital, and any terminal proceeds or recoveries. Then verify that each amount appears at the time it is paid or received.
- Discounting net income. After-tax operating income still includes depreciation as an expense. Add back depreciation when converting it to operating cash flow.
- Subtracting depreciation as if it were a cash payment. Depreciation is noncash. Its cash-flow effect in this model is through the tax reduction, not through a payment equal to depreciation.
- Omitting the initial investment. The cash purchase price is an outflow at the date it is paid. Depreciating the asset over time does not remove that outflow from NPV.
- Counting both an initial outlay and an accounting expense for the same purchase. The initial capital expenditure is included as cash at time 0; depreciation is used to calculate taxable income in later periods. Do not subtract the purchase price again as an operating expense.
- Ignoring timing of working capital. Include the investment when funds are tied up and any recovery when it is received. A recovery at the end of the project must be discounted from that date.
- Using the simple tax formula without checking its assumptions. The formula assumes positive taxable income and immediate use of the depreciation deduction. Losses, delayed deductions, or other tax rules can change the tax cash flows.
A concise check is to compare the annual cash flow with the accounting calculation. Under the stated assumptions, operating cash flow must equal after-tax operating income plus depreciation. The NPV timeline must also show the actual capital outlay and any other cash investment separately. If either check fails, the calculation is not yet a project cash-flow valuation.
Check Your Understanding
For each problem, assume annual cash flows occur at year-end, the stated required return is effective annually, and there are no tax complications beyond those specified. Amounts stated in thousands are in thousands of dollars.
- A project costs $40,000 at time 0 and generates annual cash operating contribution of $30,000 for two years. Depreciation is straight-line to zero, taxable income is positive, the tax rate is 20%, and the required return is 10%. Find NPV to the nearest cent. (Answer: $8,595.04)
- A project costs $90,000 and generates annual cash operating contribution of $45,000 for three years. Depreciation is straight-line to zero over three years, the tax rate is 30%, and the required return is 10%. Taxable income is positive. Find NPV to the nearest cent. (Answer: $10,717.51)
- A four-year project requires equipment costing $100,000 and has no other outlay or terminal cash flow. Annual cash operating contribution is $55,000; depreciation is straight-line to zero; the tax rate is 20%; and the required return is 8%. Find NPV to the nearest cent. (Answer: $62,294.22)
- A project requires equipment costing $60,000 and working capital of $5,000 at time 0. It generates annual cash operating contribution of $28,000 for three years. Depreciation is straight-line to zero over three years, taxable income is positive, and the tax rate is 25%. The $5,000 working-capital investment is fully recovered at the end of year 3. At a 10% effective annual required return, find NPV to the nearest cent. (Answer: $3,414.73)