The formula in plain words
NPV adds up what a project is worth today. You take each future cash flow, divide it by 1 plus the discount rate raised to the number of years until you receive it, and add the results together. Then you subtract what you pay at the start. The discount rate is the return you require for taking on this much risk.
NPV = CF0 + CF1/(1+r) + CF2/(1+r)^2 + ... + CFn/(1+r)^n, where CF0 is the amount you pay today, entered as a negative number.
A worked example
A company can buy a machine for $250,000 today. It expects the machine to bring in $80,000, $95,000, $110,000, and $60,000 at the end of years 1 through 4. The company requires a 9% return on projects like this one. Is the machine worth buying?
| Year | Cash flow | Discount factor | Present value |
|---|---|---|---|
| 0 | -$250,000 | 1.0000 | -$250,000.00 |
| 1 | $80,000 | 0.9174 | $73,394.50 |
| 2 | $95,000 | 0.8417 | $79,959.60 |
| 3 | $110,000 | 0.7722 | $84,940.18 |
| 4 | $60,000 | 0.7084 | $42,505.51 |
| NPV | $30,799.79 |
The four present values add up to $280,799.79. Subtract the $250,000 paid today and the NPV is $30,799.79. It is positive, so at a 9% required return the machine is worth buying.
Check it against IRR and payback
| Required return | NPV | Decision |
|---|---|---|
| 9% | $30,799.79 | Accept |
| 12% | $13,588.90 | Accept |
| 15% | -$1,969.15 | Reject |
NPV falls as the required return rises. It crosses zero between 12% and 15%, and the exact crossing point is the internal rate of return, which here is 14.60%. Because 14.60% is above the 9% requirement, IRR and NPV agree. When projects compete with each other, or when the cash flows change sign more than once, the two can disagree, and NPV is the one to trust.
Payback for this machine is about 2.68 years. The cumulative cash flow is still negative at the end of year 2 (-$75,000) and positive at the end of year 3 (+$35,000), so payback lands partway through year 3. Payback is quick to compute, but it ignores the time value of money and any cash that arrives after the payback point.
The same five steps every time
- Put the cash flows on a timeline. Cash you pay is negative and cash you receive is positive.
- Decide the discount rate. It should match the risk of this project, not the interest rate on a loan.
- Discount each cash flow by the number of years until it arrives. Year 0 is not discounted.
- Add everything up, including the negative cost at the start.
- Compare to zero. Above zero accept, below zero reject, and exactly zero means you earn the required return and no more.
Mistakes that cost points on exams
- Discounting the initial cost. Money you pay today is already in today's dollars, so year 0 gets a factor of 1.
- Using the wrong rate, such as the interest on a loan or a company-wide rate for a project with different risk.
- Losing the negative sign on the initial investment, which makes every project look wonderful.
- Counting sunk costs. Money already spent does not change the decision and does not belong in the cash flows.
- Mixing timing. Most textbook problems put each cash flow at the end of the year unless the question says otherwise.