Net Present Value (NPV)

Net present value is the present value of a project’s expected cash inflows less the present value of its expected cash outflows, with each future flow discounted at the return the business requires on the money it ties up. A positive NPV means the project clears that required return and adds value on top of it, and a negative NPV says reject.

The NPV Formula and Its Terms

         n      CFt
NPV  =   Σ   ----------   -   C0
        t=1   (1 + r)^t

CFt is the net cash flow expected in period t, that period’s inflows less its outflows. The rate r is the required return for one period, the discount rate. The counter t is the period number, n is the last period in the appraisal, and C0 is the cash paid out at the start.

Two rules govern what may enter the numerator. Only relevant cash flows belong, and to be a relevant cash flow an item must be a future, incremental cash flow, which rules out sunk costs, apportioned or allocated fixed costs, and notional costs such as depreciation (ACCA, Investment appraisal). Financing costs are left out too, because the cost of financing is accounted for in the discount rate used (ACCA, Investment appraisal). Put interest in the cash flows and then discount at a rate that already contains it, and the project pays for its funding twice.

The rate itself is the return the providers of capital require, which cost of capital sets out in full, and the equity leg of that blend is usually priced through the CAPM equation.

The Sign Convention on the Outlay

Cash paid at the start is not discounted. T0 is now and T1 is in one year’s time, so a project that starts immediately has its outlay sitting at T0, already in present-value terms, while every later flow gets divided down (ACCA, Investment appraisal).

The sign is what people get wrong. The outlay enters as a negative amount, or equivalently is subtracted from the discounted inflows, and entering it positive turns a rejection into an acceptance without changing a single input.

Spreadsheets carry their own version of the trap. Excel’s NPV function begins the investment one period before the date of the value1 cash flow and ends with the last cash flow in the list, so where the first cash flow occurs at the beginning of the first period, that first value must be added to the NPV result and not included in the values arguments (Microsoft, NPV function). Passing the outlay in as the first value shifts every flow one period later than it actually occurs, so the whole answer comes out divided by one plus the discount rate, which pulls it toward zero.

A Worked Example

The inputs below are stated for the example rather than drawn from a real project.

Initial outlay at T0                  120,000   stated input
Net cash inflow, years 1 to 3          50,000   stated input, each year
Required return                           10%   stated input

Discount factors
  T1   1 / 1.10                      =  0.9091
  T2   1 / 1.10^2                    =  0.8264
  T3   1 / 1.10^3                    =  0.7513

Present values
  T1   50,000 x 0.9091               =  45,455
  T2   50,000 x 0.8264               =  41,320
  T3   50,000 x 0.7513               =  37,565
                                        -------
  Present value of inflows              124,340
  Less outlay at T0                    (120,000)
                                        -------
  NPV                                     4,340

The result is positive, so the project earns more than the return it was asked to earn and the rule says accept. Raise the required return and the same cash flows give a smaller answer, because a higher discount rate shrinks every flow that arrives later.

Where the Number Gets Produced

In a firm this is advisory work, not compliance work. It comes up when a client is weighing equipment, a lease, a new location or a system, and when the firm faces the same choices in its own practice. The forecast is built from the project’s own incremental flows, the operating cash it brings in less the cash it absorbs, period by period. Two other appraisal measures answer different questions about the same project: the payback period reports how long the money is exposed, and its discounted form reports the point at which net present value turns positive, while the accounting rate of return works on profits instead of cash.

Net Present Value Is Not the Internal Rate of Return

Both discount the same cash flows, and they answer in different units. Net present value answers in money, stating how much a project is worth beyond the return required on it. The internal rate of return answers in percent, and it is defined as the discount rate which, when applied to the cash flows of a project, produces a net present value of nil (ACCA, The internal rate of return).

The gap opens on ranking. With mutually exclusive projects the IRR method can result in the wrong decision being made, because the project with the higher internal rate of return need not be the one with the higher net present value at the organization’s actual cost of capital, and it is the higher net present value that increases the wealth of the shareholders by the greater amount (ACCA, The internal rate of return). The rate also ignores the relative size of the investments, so a small project and a large one can carry the same percentage.

The pattern of the cash flows matters too. Where a project has non-conventional cash flows, an outflow then an inflow then another outflow, it may be possible to calculate two internal rates of return, and when a project has two there is ambiguity as to whether it should be accepted or not (ACCA, The internal rate of return). Net present value returns one figure whatever the pattern.

Common Questions

Because work like this waits behind the compliance load that fills a partner’s week, a trial task puts a block of that recurring work through Accountably’s review chain before your own reviewer depends on it.