Why Cash-Flow Signs Matter
A net present value calculation is a signed comparison: it weighs cash received against cash paid, with amounts at different dates put on a common present-value basis. The arithmetic can be straightforward, but an inconsistent sign convention can reverse the conclusion. Before entering a cash flow in an NPV equation, decide whose cash flow is being measured and whether the amount is money received or money paid by that decision-maker.
Throughout this tutorial, the perspective is that of the person or organization deciding whether to undertake a project. An amount received by that decision-maker is positive; an amount paid by that decision-maker is negative. These signs do not depend on whether a cash flow is described as a “benefit,” “cost,” “grant,” or “expense.” They depend on the direction in which money moves.
This expression assumes that cash flows occur at integer periods \(t=0,1,\ldots,n\), that \(i\) is an effective rate per period, and that the valuation date is time 0. In the examples, the periods are years, so the rate is an effective annual rate. A payment at time 0 is not discounted. If cash flows are stated from another party’s perspective, their signs may differ; do not mix the two perspectives in one equation.
Assigning Signs Before Discounting
A reliable procedure is to list the cash flows by date, give each amount a sign, combine amounts at the same date, and then discount the resulting signed flows. An amount that is paid and an amount received on the same date both belong in the equation: combine them algebraically rather than ignoring either one. A project can have a net outflow at one date and a net inflow at another.
Use the project decision-maker’s perspective throughout the calculation.
Mark receipts positive and payments negative, including costs that occur after operations end.
Add all signed cash flows occurring at the same time to obtain the net cash flow for that period.
Multiply each period’s net flow by \(v^t\), add the present values, and interpret the sign using the stated decision rule.
The sign of a present value is the sign of the cash flow itself because the discount factor \(v^t\) is positive when \(i>-1\). Discounting changes the amount’s value at the valuation date, not whether it is a receipt or a payment.
Why the Signed NPV Formula Works
The signed-sum formula follows from valuing each dated cash flow at time 0. Under a constant effective rate \(i\) per period, one unit payable at time \(t\) has present value \(v^t\). Therefore \(C_t\) units payable at that date have present value \(C_t v^t\). This applies to negative as well as positive amounts: a payment of \(K\) units is the cash flow \(-K\), with present value \(-K v^t\).
The project’s net present value is the sum of the present values of all its receipts and payments. The signed sum is therefore not a special adjustment applied after finding the project’s benefits and costs; it is the direct valuation of the project’s net cash flows. For example, a receipt and a payment at the same date have present values \(R v^t\) and \(-K v^t\), whose sum is \((R-K)v^t\). Combining them first gives the same result.
The NPV sign does not by itself establish that every possible decision-maker should choose the project. The rule above is conditional on the stated decision criterion. If the problem gives a different rule, apply that rule rather than silently substituting one.
Worked Example: A Grant, a Cost, and Annual Receipts
A project requires a $9,000 equipment payment at time 0, and the decision-maker receives a $1,000 grant at the same time. It produces receipts of $3,200 at the end of year 1, $3,500 at the end of year 2, and $4,000 at the end of year 3. A $500 maintenance payment is also due at the end of year 2. The effective annual interest rate is 6%. Find the NPV and state the result under a rule that accepts the project only if NPV is positive.
Method. Assign signs from the project decision-maker’s perspective, combine flows at each date, and discount each annual net flow at 6%.
At time 0, the equipment payment is \(-9{,}000\) and the grant is \(+1{,}000\), giving a net flow of \(-8{,}000\). At year 1 the net flow is \(+3{,}200\). At year 2 the receipt and maintenance payment give \(3{,}500-500=3{,}000\). At year 3 the net flow is \(+4{,}000\).
Rounded to the nearest cent, the NPV is $1,046.98. It is positive, so the project meets the stated positive-NPV acceptance rule. The calculation avoids the common mistake of treating the grant as a payment or forgetting to subtract the maintenance cost from the year-2 receipt.
Worked Example: A Closure Cost Reverses the Conclusion
A proposed facility costs $6,000 at time 0. It produces receipts of $2,500 at the end of year 1 and $3,000 at the end of year 2. At the end of year 3, it generates a receipt of $1,800, but the decision-maker must also pay $1,000 to close the facility. The effective annual interest rate is 8%. Calculate NPV and decide whether the project passes a rule requiring positive NPV.
Method. Treat the closure expense as an outflow at year 3, not as an additional receipt. The year-3 net cash flow is \(1{,}800-1{,}000=800\).
Rounded to the nearest cent, NPV is -\$478.10, so it fails the stated rule. A sign error can produce the opposite decision. If the closure payment is incorrectly treated as positive, the year-3 flow becomes \(1{,}800+1{,}000=2{,}800\), and the incorrect NPV is
That incorrect result is positive because it counts a required payment as money received. The correction is to give the closure cost a negative sign before combining the year-3 amounts.
Worked Example: Mixed Flows at Every Stage
A hypothetical recycling project has a $5,000 purchase payment and an $800 setup grant at time 0. At the end of year 1, it receives $2,200 and pays $300 in operating expenses. At the end of year 2, it receives $3,000 and pays $700 for maintenance. At the end of year 3, it receives $600 for equipment resale and pays $1,000 for shutdown. The effective annual rate is 10%. Find NPV and interpret it using a positive-NPV acceptance rule.
Method. Work out the signed net cash flow at each date, then discount those net amounts. The time-0 net flow is \(-5{,}000+800=-4{,}200\). The year-1 net flow is \(2{,}200-300=1{,}900\). The year-2 net flow is \(3{,}000-700=2{,}300\). The year-3 net flow is \(600-1{,}000=-400\); the final year is a net outflow despite the resale receipt.
Rounded to the nearest cent, NPV is -\$872.43, so the project does not pass the stated rule. The negative year-3 amount must remain negative when discounted. A common mistake is to focus on the resale proceeds and omit the larger shutdown payment, incorrectly making the final year look like a net receipt.
Common Sign Errors and Checks
When an NPV answer seems counterintuitive, review the signed cash-flow list before repeating the discounting. The following checks catch frequent errors:
- Keep one perspective. A payment by the project is an outflow for the project decision-maker. Do not switch midway to the perspective of a supplier, lender, or grant provider.
- Do not make all listed amounts positive. A problem may list receipts and costs as unsigned dollar amounts; you must assign their signs from the descriptions.
- Include costs at the end. Closure, removal, or other required payments remain outflows even when they occur after the main operating period.
- Combine same-date flows algebraically. A receipt does not erase a cost unless their signed amounts actually offset it.
- Leave time-0 cash flows undiscounted. Their discount factor is \(v^0=1\), so the time-0 net amount enters the NPV unchanged.
As a quick reasonableness check, verify that the NPV’s sign is consistent with the discounted totals: positive NPV means the signed present values of inflows exceed those of outflows; negative NPV means they do not. This check does not replace the calculation, but it can reveal a reversed sign or a missing expense.
Check Your Understanding
For each problem, use the decision-maker’s perspective and round money to the nearest cent. Assume all stated rates are effective annual rates and the flows occur at the ends of the stated years.
- A project costs $4,000 at time 0 and returns $2,200 at the end of year 1 and $2,500 at the end of year 2. At 5%, find NPV. (Answer: $362.81)
- At time 0, a machine costs $7,500 and a $500 rebate is received. It generates receipts of $3,000 at the end of year 1 and $4,000 at the end of year 2. At 10%, find NPV and state whether it is positive. (Answer: -$966.94; negative)
- A project has a time-0 equipment payment of $6,800, a time-0 deposit payment of $1,200, and a time-0 rebate of $1,000. At the end of year 1, receipts are $3,400 and operating expenses are $600. At the end of year 2, receipts are $5,000 and a cleanup expense is $1,500. At 8%, find NPV. (Answer: -$1,406.72)
- A project costs $9,500 at time 0 and receives a $1,500 grant at that time. It has a net receipt of $1,500 at the end of year 1, a $4,000 receipt at the end of year 2, and a $5,000 receipt plus a $1,200 closure payment at the end of year 3. At 6%, find NPV and apply a rule that accepts projects only when NPV is positive. (Answer: $165.63; accept under the stated rule)