# Net present value: the discounted cash flow formula for evaluating projects
TL;DR
- Net present value discounts each future cash flow to today's dollars and subtracts the initial outlay, using NPV = Σ [CF_t ÷ (1 + r)^t] − C₀. - A positive NPV indicates a project modeled to earn more than the chosen discount rate, while a negative NPV indicates it falls short of that benchmark. - A $100,000 outlay with five rising cash flows models to an NPV of about +$29,079 at a 10% discount rate and turns slightly negative near 19.7%, the internal rate of return. - The discount rate and the cash flow forecast drive nearly all of the result, so a sensitivity table is more informative than a single point estimate.
Capital budgeting questions share one structure: money leaves now, and money arrives later in uncertain amounts. A raw sum of those future amounts overstates their worth, because a dollar received in year five cannot be put to work for the four years before it arrives. Net present value resolves the timing problem by converting every cash flow to a common date, typically today, so that a project's inflows and outflows can be compared on equal terms.
Informational calculation reference only.
All equations, tools, and outputs on this page are intended strictly for educational modeling and mathematical illustration. They do not constitute certified financial, legal, or tax advice. For specific scenarios, consult a certified public accountant (CPA) or a fiduciary financial advisor.
The mathematical formula behind the calculation
The present value of a single future cash flow $CF_t$ received $t$ years from now, at a discount rate $r$, is:
PV = \frac{CF_t}{(1 + r)^{t}}
Net present value sums the present values of every period's cash flow and subtracts the initial investment $C_0$ made at time zero:
NPV = \sum_{t=1}^{n} \frac{CF_t}{(1 + r)^{t}} - C_0
Two companion metrics are derived from the same discounted values. The profitability index expresses present value of inflows per dollar of initial outlay:
PI = \frac{\sum_{t=1}^{n} \frac{CF_t}{(1 + r)^{t}}}{C_0}
The internal rate of return is the discount rate at which NPV equals exactly zero:
0 = \sum_{t=1}^{n} \frac{CF_t}{(1 + IRR)^{t}} - C_0
The inputs are summarized below.
| Symbol | Definition | Modeling note |
|---|---|---|
| $CF_t$ | Net after-tax cash flow in period $t$ | Cash, not accounting profit |
| $r$ | Discount rate | Often a cost of capital or alternative return of similar risk |
| $t$ | Period number | Annual periods in the standard layout |
| $C_0$ | Initial investment at time zero | Entered as a positive outlay |
| $n$ | Number of forecast periods | Terminal value is modeled separately |
Step-by-step practical calculation example
Consider a project that requires a $100,000 outlay today and is forecast to return $25,000, $30,000, $35,000, $40,000, and $45,000 in years one through five. The assumed discount rate is 10%.
Step 1: compute each discount factor. The factor for year $t$ is $(1.10)^t$, which gives 1.1000, 1.2100, 1.3310, 1.4641, and 1.6105.
Step 2: discount each cash flow.
| Year | Cash flow | Discount factor | Present value | Cumulative present value |
|---|---|---|---|---|
| 1 | $25,000 | 1.1000 | $22,727 | $22,727 |
| 2 | $30,000 | 1.2100 | $24,793 | $47,521 |
| 3 | $35,000 | 1.3310 | $26,296 | $73,817 |
| 4 | $40,000 | 1.4641 | $27,321 | $101,137 |
| 5 | $45,000 | 1.6105 | $27,941 | $129,079 |
$$ NPV = 129{,}079 - 100{,}000 = +\$29{,}079 $$
Step 4: derive the companion metrics. The profitability index is $129{,}079 \div 100{,}000 = 1.29$, meaning about $1.29 of present value per $1.00 committed. The cumulative column shows that discounted inflows pass $100,000 during year four, so the discounted payback period is about $3 + (100{,}000 - 73{,}817) \div 27{,}321 \approx 3.96$ years. Undiscounted, the same cash flows recover the outlay in about 3.25 years, which illustrates why simple payback understates the time required once the time value of money is applied.
The positive result does not mean the project produces $29,079 of profit in the ordinary sense. Under the stated assumptions, it means the project is modeled to deliver the full 10% required return on the capital employed, plus a surplus of $29,079 expressed in today's dollars.
Strategic applications for financial modeling
Comparing mutually exclusive projects. When two projects compete for the same capital, NPV measures value created in dollars at a common discount rate. A smaller project with a higher IRR can still produce a lower NPV than a larger project, so the two metrics can rank the same options differently.
Capital rationing. When funds are limited, the profitability index ranks projects by present value created per dollar invested. A project with an NPV of $29,000 on a $100,000 outlay and another with an NPV of $20,000 on a $40,000 outlay show how the index reorders a list that absolute NPV would rank differently.
Sensitivity analysis. Because the result depends heavily on the discount rate, modeling a range is more informative than a single figure. Using the same cash flows:
| Discount rate | Modeled NPV |
|---|---|
| 6% | +$44,982 |
| 8% | +$36,680 |
| 10% | +$29,079 |
| 12% | +$22,104 |
| 15% | +$12,680 |
| 20% | −$704 |
Cash flow forecast stress tests. Scaling every cash flow by 90% in the example above lowers the modeled NPV at 10% from about $29,079 to about $16,171, a decline of roughly 44% from a 10% forecast reduction. Because the initial outlay is fixed, a modest forecast shortfall is magnified in the result, which is why the quality of the forecast typically matters more than the third decimal of the discount rate.
Business valuation. The same arithmetic, applied to a company's projected free cash flows plus a terminal value, underlies discounted cash flow valuation. The structure is identical; only the horizon and the terminal assumption change.
Common pitfalls and variable mistakes
1. Using accounting profit instead of cash flow. Depreciation is a non-cash expense and is added back, while capital expenditures and increases in working capital are subtracted. Mixing the two distorts the series. 2. Double-counting financing costs. When the discount rate is a weighted average cost of capital, interest payments are excluded from the cash flows, since the rate already reflects the cost of debt. 3. Ignoring taxes. Cash flows are generally modeled after tax. A pre-tax series discounted at an after-tax rate overstates value. 4. Applying one rate to risks of different size. A discount rate typically rises with the uncertainty of the cash flows. A rate borrowed from a lower-risk project understates the hurdle. 5. Relying on IRR alone. Cash flows that change sign more than once can produce multiple valid IRRs, and IRR implicitly assumes interim cash flows are reinvested at the IRR itself. NPV does not carry either limitation, which is why it is commonly treated as the more reliable rule when the two disagree. 6. Forgetting the timing convention. The standard layout discounts each flow as if it arrives at year end. Flows that arrive evenly during the year would model to a somewhat higher present value under mid-year discounting. 7. Omitting terminal value or salvage value. Truncating the horizon at the end of the forecast can understate a long-lived asset's value, while an aggressive terminal assumption can dominate the result. The share of total present value that comes from the terminal term is worth reporting separately.
Ranges used for discount rates vary by context. Mature corporations often model somewhere around 8% to 12%, small businesses commonly 10% to 15%, and early-stage ventures considerably higher, reflecting the greater dispersion of outcomes. These figures are illustrative modeling ranges rather than recommendations.
To enter a cash flow series, compute NPV, IRR, the profitability index, and discounted payback, and view a discount rate sensitivity table, use the NPV calculator.
Disclaimer: While we strive for absolute mathematical precision, actual real-world financial outcomes may vary based on institutional fees, localized tax brackets, changes in federal legislation, or fluctuating market indexes.