Lesson 4.2 40 min
Net present value
Which cash flows to include, how to discount them, and why a positive NPV is the test for adding value.
Net present value, or NPV, adds up the present value of every cash flow from a project, outflows included, at a stated discount rate.
NPV = Cash flow at year 0 + Cash flow at year 1 ÷ (1 + r) + Cash flow at year 2 ÷ (1 + r)^2 + ...
The decision rule is short. If NPV is above zero, the project returns more than the discount rate and adds value. If it is below zero, it returns less. A project with an NPV of exactly zero earns the required rate and nothing more.
Which cash flows count
NPV uses cash flows, not accounting profit. Four rules decide what goes in.
- Include only incremental cash flows: the difference between the business with the project and without it. Revenue the second espresso machine takes from the first machine does not count.
- Leave out depreciation. It is an expense with no cash movement. The cash went out when the machine was bought, and that is where it belongs.
- Leave out sunk costs. A design fee already paid, or a survey already done, is the same whether or not the project goes ahead.
- Include opportunity costs. If the project uses a shop corner that could otherwise be rented for $2,000 a year, that rent is a cost of the project.
Add the sale value of equipment at the end, any working capital the project ties up (and its release at the end), and the effect of tax. The case below ignores tax and working capital so the method is easy to follow. A real appraisal needs both. Lesson 5.1 shows how working capital behaves.
Timing
Cash at year 0 is not discounted. Cash arriving at the end of year 1 is discounted one period. The case uses annual end-of-year flows, which is a simplification: real cash arrives through the year.
Choosing the discount rate
The rate has to reflect the risk of the project's cash flows. A rate set at the bank's loan rate treats the project as riskless, since the bank is paid before the owners. Most small businesses add a margin for risk to their cost of borrowing. The margin is a judgement. The way to handle uncertainty about it is to try several rates and see where the decision changes, which the case does below.
The spreadsheet trap
Excel and other spreadsheets have an NPV function. It assumes the first cash flow in its list comes at the end of the first period. Microsoft's documentation says that if the first cash flow occurs at the beginning of the first period, it must be added to the function's result, not included in the range. Many NPV errors come from putting the year-0 outlay inside the function, which discounts it by a year and overstates NPV.
Worked case
Northside's second espresso machine
Northside's morning queue loses sales. A second espresso machine and grinder cost $18,000. The extra cash flows it brings are estimated at $5,200 in year 1, $6,000 in year 2 and $6,400 in each of years 3 and 4, and $6,400 in year 5 plus a resale value of $2,000. The discount rate is 10%.
| Cash flow $ | Factor at 10% | Present value $ | |
|---|---|---|---|
| Year 0 (today) | (18,000) | 1.0000 | (18,000.00) |
| Year 1 | 5,200 | 0.9091 | 4,727.27 |
| Year 2 | 6,000 | 0.8264 | 4,958.68 |
| Year 3 | 6,400 | 0.7513 | 4,808.41 |
| Year 4 | 6,400 | 0.6830 | 4,371.29 |
| Year 5 | 8,400 | 0.6209 | 5,215.74 |
| Net present value | 6,081.39 |
The inflows add to $32,400 against a $18,000 cost, so the undiscounted gain is $14,400. After discounting the gain is $6,081.39. The $2,000 resale value contributes only $1,241.84 of it, because it arrives in year 5.
Testing the discount rate:
| Discount rate | NPV $ |
|---|---|
| 8% | 7,460.46 |
| 10% | 6,081.39 |
| 12% | 4,815.12 |
| 15% | 3,102.21 |
| 20% | 665.90 |
| 25% | (1,349.25) |
NPV stays positive until the rate reaches about 21.6%, which lesson 4.3 calls the internal rate of return. It is negative at 25%, so the decision flips if Northside's required return is above that.
A second test is the cash flows. If each of years 1 to 5 comes in 15% lower and the machine has no resale value, the flows become $4,420, $5,100, $5,440, $5,440 and $5,440 and the NPV falls to $1,413.62. It is still positive, so the decision survives a weaker year than expected. A project that turns negative under a modest downgrade needs a closer look before the money is spent.
Template
NPV workbook with sensitivity
Enter a cost, up to ten years of cash flows and a rate. The sheet shows each year's discount factor and present value, the NPV by the NPV function with the year-0 outlay added outside it, and a rate table from 5% to 30%. A second sheet applies a percentage cut to the inflows. The espresso machine is loaded.
The full working file, 4-2-npv.xlsx, is in the Accounting and Finance Toolkit, Complete Track 1. The lesson above teaches the method and shows the worked numbers, so you can build it yourself in any spreadsheet.
Exercises
Work these, then open the answers
The same machine costs $14,000 instead of $18,000. What is the NPV at 10%?
Answer
The present value of the inflows is $24,081.39, so NPV = $24,081.39 − $14,000 = $10,081.39.
What is the highest price Northside could pay for the machine and still have an NPV of at least zero at 10%?
Answer
The present value of the inflows, $24,081.39.
The $2,000 of resale value is removed. Is the project still worth doing at 10%?
Answer
Removing it takes $1,241.84 off the NPV: $6,081.39 − $1,241.84 = $4,839.55. Yes, still positive.
Check
5 questions
Sources
Sources
- Principles of Finance, OpenStax, Rice University
- NPV function, Microsoft Support