The formula
NPV discounts every cash flow back to today and adds the results. Money you pay out counts as negative and money you receive counts as positive. Year 0 is whatever moves right now, usually the purchase price, so it is not discounted at all.
NPV = CF0 + CF1/(1+r) + CF2/(1+r)^2 + ... + CFn/(1+r)^n, where CFt is the cash flow in year t and r is the required return per year. This is the standard textbook definition, found for example in Berk and DeMarzo, Corporate Finance.
A worked example with uneven cash flows
A bakery can buy a second oven for $60,000. It expects extra profit of $15,000, $22,000, $28,000, $18,000 and $12,000 in years 1 through 5, and the owner wants 10% a year on money tied up in equipment. Because the amounts differ, each year gets its own discount factor, 1/(1.10)^t.
| Year | Cash flow | Factor | Present value |
|---|---|---|---|
| 0 | -$60,000.00 | 1.0000 | -$60,000.00 |
| 1 | $15,000.00 | 0.9091 | $13,636.36 |
| 2 | $22,000.00 | 0.8264 | $18,181.82 |
| 3 | $28,000.00 | 0.7513 | $21,036.81 |
| 4 | $18,000.00 | 0.6830 | $12,294.24 |
| 5 | $12,000.00 | 0.6209 | $7,451.06 |
| NPV | $12,600.29 |
The five inflows are worth $72,600.29 today, which is $12,600.29 more than the oven costs. The project clears the 10% hurdle, so accept it. Try a higher required return in the calculator: at 15% the NPV falls to $4,346.77, and at 20% it turns negative at -$2,515.43. The rate where it crosses zero is the internal rate of return, 18.06% for this oven.
Reading the answer
The sign of NPV is the decision. Above zero, the project creates value beyond the required return. Below zero, you would do better putting the money elsewhere at that return. At exactly zero you are indifferent. The size of the number is in today's dollars, so when two projects compete for the same money, the larger NPV adds more wealth.
The discount rate carries most of the judgment. Use the return investors could earn on something with the same risk, which for an average project is often the company's WACC. A riskier project deserves a higher rate. The period must also match the rate: if the cash flows are monthly, use a monthly rate.
In Excel, the NPV function assumes its first value arrives at the end of year 1, so type =NPV(rate, year 1 to year n) and then add the year 0 amount outside the brackets. A BA II Plus does the same job with the CF and NPV keys, and the guide linked below lists the keystrokes.
Mistakes that cost points
- Discounting the year 0 amount. Cash paid today is already in today's dollars and has a factor of 1.
- Feeding the whole list, including the initial cost, to Excel's NPV function. It treats the first entry as year 1, so the outlay gets discounted one year too many.
- Pairing monthly cash flows with an annual rate. Convert one of them so the units agree.
- Leaving out the last-year items that are easy to forget, such as selling the equipment or getting working capital back.
- Choosing the discount rate after seeing the answer. Set it from the risk of the project first.
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