Net present value: why the cash dates matter

A project can return more cash than it costs and still have a negative net present value at a stated discount rate. The amounts are unchanged. Their arrival dates now enter the valuation.

Net present value, or NPV, expresses dated cash inflows and outflows at one valuation date, then adds them. The OpenStax explanation discounts each future cash flow separately. The initial payment, if it occurs on the valuation date, is already there.

Two receipts and one opening payment

Consider an invented project requiring $1,000 now. It supplies $600 of net cash at the end of year one and $600 at the end of year two, then ends. These are the complete project cash flows, after its operating payments and before tax. There is no financing, resale receipt or other payment in the model.

The undiscounted result is positive $200: $600 plus $600, less $1,000. Keep that result. Now use a supplied annual discount rate of 10 percent to express the same schedule at its starting date.

Divide the first $600 by 1.10. The second arrives two years from the start, so divide it by 1.10 twice, or 1.21. The $1,000 paid now is not divided again.

One invented cash schedule, valued at its start using 10 percent annually

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The NPV is positive $41.32, calculated before rounding the displayed amounts to cents. That is a value at the starting date under the supplied assumptions. It is not a further $41.32 receipt, an annual percentage return or the company's accounting profit.

Move the money without changing the total

Try changing just one date. The first $600 still arrives at the end of year one. The second now arrives at the end of year three, with every amount and other model condition unchanged. What happens to the $200 total? What happens to NPV?

The total stays positive $200. The later receipt is now divided by 1.10 three times, giving a present value of about $450.79. Adding it to the first receipt's unrounded present value and subtracting the opening $1,000 gives an NPV of negative $3.76.

The delay has not removed any dollars from the schedule. It has changed what that schedule is worth at the start when evaluated using 10 percent. At a zero discount rate, both versions have an NPV of positive $200. The dates and the rate belong beside the result.

A higher percentage on a smaller project

Consider a separate invented choice with $1,000 available now. Only one of two projects can be undertaken. A costs $100 now and pays $130 at the end of one year. B costs $1,000 now and pays $1,200 on that same date. Both then end.

These are the complete incremental project cash flows after all project costs and before tax, with no borrowing or other payments or receipts. Use a supplied 10 percent annual discount rate for both on comparable risk assumptions.

A's $30 gain on $100 is 30 percent. B's $200 gain on $1,000 is 20 percent. For these single-outlay, single-receipt projects, those percentages are also their internal rates of return, or IRRs. The OpenStax IRR explanation defines the rate that makes a project's NPV zero and identifies project scale as a limitation of percentage rankings.

At the common 10 percent rate, A's NPV is $130 divided by 1.10, less $100: approximately $18.18. B's is $1,200 divided by 1.10, less $1,000: approximately $90.91. A has the higher percentage; B adds more value in dollars at this rate.

Keep the unused money in the comparison. The model also supplies a separate before-tax one-year alternative earning exactly 10 percent, paid at year-end, with no charges. If A is chosen, its unused $900 goes there and becomes $990. Add A's $130 receipt and the original $1,000 finishes as $1,120. Choosing B instead finishes with $1,200.

The difference is $80 at year-end. Dividing it by 1.10 gives approximately $72.73 at the starting date, matching the difference between the two unrounded project NPVs.

The comparison remains two one-off projects. Repeating A ten times would require ten available projects, an option this model has not supplied. Keep the amount committed, the dated receipts, the shared discount rate and the use of unused money beside a return percentage.

Two rates can make the same project zero

An internal rate of return need not be unique. Consider a separate invented project with three complete cash flows: pay $12,000 now, receive $32,400 at the end of year one, then pay a required $21,600 closure cost at the end of year two. The flows are after all project operating costs and before tax, with no financing, other receipts or other payments.

The schedule changes from an outflow to an inflow and back to an outflow. ACCA's 2018 financial-management sample answers identify multiple IRRs as a possible problem when project cash flows reverse direction more than once. It is a possibility, not a claim that every such schedule has multiple usable answers.

For this supplied schedule, both 20 percent and 50 percent annually make NPV zero:

• At 20 percent: $32,400 divided by 1.20 is $27,000. The final $21,600 payment divided by 1.20 squared is $15,000. The $27,000 receipt value, less $12,000 now and $15,000 for closure, gives zero.
• At 50 percent: the receipt's present value is $21,600 and the final payment's is $9,600. Subtract $12,000 and $9,600 from $21,600: zero again.

Neither calculation has changed a cash flow. Selecting the larger rate would not establish that the project earns a single unambiguous 50 percent return. A calculation that finds one zero has not demonstrated that no other zero exists.

Use a separately supplied comparison rate to examine NPV. At 30 percent annually, this schedule's NPV is approximately positive $142.01. At 10 percent it is approximately negative $396.69. Both use the same three payments and their original dates, calculated before rounding. These are illustrative rates, not recommendations or available financing offers.

The undiscounted total is negative $1,200: $32,400 less $12,000 and $21,600. A positive NPV at the supplied 30 percent rate does not turn that total into a cash gain. The early receipt and later payment make the timing consequential. The closure payment also remains due. Its funding needs a separate dated plan; an NPV result does not arrange it.

When reading an IRR result, recover the complete signed cash schedule first. If later outflows follow receipts, check whether one reported rate can answer the decision at all. Do not choose the preferred answer by changing the payment signs or omitting the closing obligation.

Read the inputs before the answer

The 10 percent here is a supplied comparison input, not a claimed market rate or a promise that the project earns 10 percent. A different rate changes the answer. So would a missing closing payment, an overstated receipt or a different receipt date. Discounting an unsupported forecast does not supply its missing evidence.

Nor does a positive NPV put the opening $1,000 into the bank. A dated cash forecast examines whether the payments can be funded. Simple payback follows the recovery date. NPV keeps all the modeled cash flows and their dates in the valuation; it does not replace those other readings.

Before repeating an NPV figure, recover its cash schedule, valuation date and discount rate. Then change a consequential date and see whether the conclusion survives. To read the business decisions behind Infinite Upside: Building Wealth With Room to Live, start with the opening.