How do I calculate project net present value?

People type this into search every day: how do I calculate project net present value.

Net present value discounts each year’s cash inflow, adds those present values, and subtracts the outlay. Investor.gov writes compound interest as future value = present value × (1 + r) ^ n, on the compound interest glossary page. The project NPV calculator runs that backward on a repeating annual inflow.

PV of one inflow = cash ÷ (1 + r) ^ year

PV of the series = cash × [1 − (1 + r) ^ (−n)] ÷ r

NPV = that present value − outlay

At a 0% rate the present value is cash × years. The inflow is the same amount at the end of each year. Tax, a residual sale, and a second outlay sit outside the repeating-cash formula. You can fold a residual in by hand, and example 9 does that. These are labeled examples. They are not a funding decision.

The present value of a future sum page discounts one amount. This page discounts a run of equal amounts and then subtracts what you spend on day one. Simple payback skips the discount.

Example 1. $50,000 in, $15,000 a year for five years, at 8%

Calculator defaults:

• Project outlay: $50,000
• Annual net cash inflow: $15,000
• Years of inflows: 5
• Annual discount rate: 8%

Undiscounted inflows: 15,000 × 5 = $75,000.00.

Present value of each year’s $15,000:

• Year 1: 15,000 ÷ 1.08 = $13,888.89
• Year 2: 15,000 ÷ 1.08 ^ 2 = $12,860.08
• Year 3: 15,000 ÷ 1.08 ^ 3 = $11,907.48
• Year 4: 15,000 ÷ 1.08 ^ 4 = $11,025.45
• Year 5: 15,000 ÷ 1.08 ^ 5 = $10,208.75

Sum: $59,890.65. That matches the annuity form, 15,000 × [1 − 1.08 ^ (−5)] ÷ 0.08.

NPV: 59,890.65 − 50,000 = $9,890.65.

The time-value haircut is 75,000 − 59,890.65 = $15,109.35. You still clear the outlay at 8%. The leftover is $9,890.65 in today’s dollars.

Cumulative present value: year 1 $13,888.89, year 2 $26,748.97, year 3 $38,656.45, year 4 $49,681.90, year 5 $59,890.65. The line crosses $50,000 during year 5. A 4-year life is example 6, and it finishes $318.10 short.

Example 2. The same project at a 0% discount rate

Rate: 0%. Outlay, cash, and years stay at the defaults.

Present value: 15,000 × 5 = $75,000.00.

NPV: 75,000 − 50,000 = $25,000.00.

At 0% there is no haircut. NPV equals the undiscounted leftover. The $25,000 is a real output of the calculator when you type 0%. It is also the number people sometimes call NPV while the rate box still says 8%. Example 8 separates those two.

Example 3. What if the rate is 4%, or 12%?

Same $50,000, $15,000, and five years.

At 4%, the present value of the inflows is $66,777.33. NPV is 66,777.33 − 50,000 = $16,777.33.

At 12%, the present value is $54,071.64. NPV is $4,071.64.

A higher rate says a later dollar is worth less, because the alternative you typed pays more. Moving from 4% to 12% takes $12,705.69 off the NPV (16,777.33 − 4,071.64). The cash you typed did not change. The rate did.

If the $15,000 is already in today’s dollars, a rate that also includes inflation discounts the cash a second time and the NPV comes out lower. Keep the cash and the rate on one basis. The two results above are the size of that choice on this project: $16,777.33 at 4%, and $4,071.64 at 12%.

Example 4. Around the project’s own rate, and past it

The IRR on this default project is 15.2%. NPV at a few rates nearby:

At 15%, present value is $50,282.33. NPV is $282.33.

At 15.2%, present value is $50,045.14. NPV is $45.14.

At 20%, present value is $44,859.18. NPV is −$5,140.82, printed as -$5,140.82. The label is negative net present value.

The printed IRR is rounded to one decimal. Typing 15.2% back into the NPV page leaves $45.14, because the rate that zeros NPV sits a little above 15.2%. At 20% the discounted inflows no longer cover $50,000.

Example 5. What if the inflows run for ten years?

Same $50,000, same $15,000 a year, same 8%. Years: 10.

Undiscounted inflows: 15,000 × 10 = $150,000.00.

Present value: $100,651.22.

NPV: 100,651.22 − 50,000 = $50,651.22.

The extra five years add 50,651.22 − 9,890.65 = $40,760.57 of NPV. Those later years are discounted harder, so they add less than the first five years’ $9,890.65 plus another copy of it. The first five years contributed $59,890.65 of present value. The second five add 100,651.22 − 59,890.65 = $40,760.57 of present value, which is the whole of the extra NPV.

Example 6. What if the project only runs three years, or four?

Simple payback on $50,000 at $15,000 a year is 50,000 ÷ 15,000 = 3.33 years. A life shorter than that cannot cover the outlay even before discounting.

Three years at 8%. Undiscounted inflows: $45,000.00. Present value: $38,656.45. NPV: 38,656.45 − 50,000 = −$11,343.55, printed as -$11,343.55.

Four years at 8%. Undiscounted inflows: $60,000.00. Present value: $49,681.90. NPV: −$318.10, printed as -$318.10.

Four years of cash add to $60,000, which is $10,000 more than the outlay, and the NPV is still negative. The $10,000 arrives late, and at 8% it is not enough to close a $318 gap. Year 5 is the year the default project crosses, in example 1.

Example 7. What if each year pays $20,000?

• Outlay: $50,000
• Annual cash: $20,000
• Years: 5
• Rate: 8%

Undiscounted inflows: $100,000.00.

Present value: $79,854.20.

NPV: 79,854.20 − 50,000 = $29,854.20.

The extra $5,000 a year is 20,000 ÷ 15,000 − 1 = 33.3% more cash. NPV rises from $9,890.65 to $29,854.20. That is 29,854.20 ÷ 9,890.65 = 3.02 times the leftover, because the outlay is a fixed subtraction and the extra cash is discounted on top of a series that had already cleared it.

Example 8. Calling the undiscounted leftover the NPV

Defaults again: five inflows, $75,000 in total, $50,000 outlay, 8% rate.

Undiscounted leftover: 75,000 − 50,000 = $25,000.00.

That $25,000 is the example 2 result, the NPV at a 0% rate.

NPV at the 8% you typed is $9,890.65.

The gap between those two readings is the $15,109.35 haircut from example 1. If a spreadsheet shows $25,000 and the rate cell says 8%, the rate cell is not in the sum. The year-by-year division in example 1 is the check.

A larger outlay makes the same present value fail:

• Outlay: $80,000
• Annual cash: $15,000
• Years: 5
• Rate: 8%

Present value of inflows: still $59,890.65.

NPV: 59,890.65 − 80,000 = −$20,109.35, printed as -$20,109.35.

The inflows did not change. Another $30,000 of outlay turns a $9,890.65 surplus into a $20,109.35 deficit. 9,890.65 − 30,000 = −20,109.35.

Example 9. One future sum, then a residual sale in year 5

The present value calculator discounts a single amount.

PV = future amount ÷ (1 + rate) ^ years

Defaults: $10,000 in 10 years at 5%.

Growth factor: 1.05 ^ 10 = 1.63×.

Present value: 10,000 ÷ 1.62889… = $6,139.13.

Amount discounted: 10,000 − 6,139.13 = $3,860.87.

The NPV page cannot take a different last year on its own. A residual sale gets added by this one-sum formula. Sell the asset for $10,000 at the end of year 5, and discount that sale at the same 8% as example 1.

Present value of the sale: 10,000 ÷ 1.08 ^ 5 = $6,805.83.

Amount discounted on the sale: $3,194.17. Growth factor: 1.47×.

NPV of the repeating $15,000 plus the sale: 9,890.65 + 6,805.83 = $16,696.48.

The repeating series and the one-time sale stay on the same rate. Mixing an 8% series with a 5% sale would be two different alternatives in one total.

Example 10. A second outlay in year 3

The repeating-cash page assumes cash goes out once. A project that spends again later needs a year list. Use the same Investor.gov discount, one year at a time, at 8%.

• Year 0: −$50,000, present value −$50,000.00
• Year 1: $15,000, present value $13,888.89
• Year 2: $15,000, present value $12,860.08
• Year 3: $15,000 of inflow and a $20,000 extra spend, so −$5,000. Present value: −5,000 ÷ 1.08 ^ 3 = −$3,969.16
• Year 4: $15,000, present value $11,025.45
• Year 5: $15,000, present value $10,208.75

NPV: −50,000 + 13,888.89 + 12,860.08 − 3,969.16 + 11,025.45 + 10,208.75 = −$5,985.99.

The level project in example 1 had an NPV of $9,890.65. The extra $20,000 in year 3, net of that year’s $15,000 inflow, moves the result by 9,890.65 − (−5,985.99) = $15,876.64. The IRR page assumes one sign change, cash out once and then inflows. Two sign changes can produce more than one rate. On this shape I would keep the −$5,985.99 NPV at the 8% hurdle and skip the search for a single IRR.

Ten NPV cases side by side

Open the NPV calculator with the outlay, the yearly cash, the year count, and a rate you are willing to compare against. The rate that drives this NPV to zero is the internal rate of return. A single sum, with no yearly repeat, belongs on the present value calculator.