NPV, IRR and Payback Calculator

Cash flows (outflows negative, inflows positive; a block copied from a spreadsheet column can be pasted into any box)

Same amount in several periods

How to use it
  1. Enter the discount rate per period, usually the minimum attractive rate of return (MARR).
  2. Enter the cash flow in each period, starting with period 0 (today). Enter costs as negative and savings or income as positive. Leave a period blank if it has no cash flow.
  3. Use “Add a period” for more periods, up to 40.
  4. For MIRR, enter the finance rate (what the outflows cost) and the reinvestment rate (what the inflows earn). Left blank, both are the discount rate.
  5. NPV, IRR, MIRR, payback and the other results appear and update as you type.

To see it filled in, choose “Show an example”. The rate and the periods must match: a 6 % annual rate with yearly cash flows, or 0.5 % a month with monthly ones.

Result

Enter the discount rate.

Each cash flow falls at the end of its period. Payback is interpolated within the period in which the running total turns positive, as if that period’s cash flow arrived evenly through it.

Methodology, equations and example

This calculator evaluates a series of cash flows, one per period, with the standard discounted cash flow measures of engineering economics and corporate finance. Period 0 is today, and each later cash flow falls at the end of its period. It follows the notation of typical engineering-economy texts (e.g., Blank & Tarquin; Newnan). For comparing alternatives; not financial advice.

Variables

  • CFt – Cash flow in period t, negative for an outflow and positive for an inflow
  • n – Number of periods, the last period with a cash flow
  • i – Discount rate per period, usually the minimum attractive rate of return (MARR): the lowest rate of return the owner will accept on the money

Net present value (NPV)

NPV = Σ CFt / (1 + i)t, for t = 0 to n

NPV is the value today of every cash flow, each discounted at i. A positive NPV means the cash flows return more than the discount rate, so the project adds value at that rate. Among mutually exclusive alternatives with equal lives, the one with the highest NPV is the best choice.

Internal rate of return (IRR)

The IRR is the discount rate at which NPV = 0. There is no closed form, so the calculator scans NPV across rates from −99 % to 10,000 % per period and closes in on each change of sign by bisection. When the IRR is above the discount rate, the NPV is positive, and the two measures agree on whether a single project is worthwhile.

A conventional project has one change of sign in its cash flows (costs first, then savings), and so one IRR. By Descartes' rule of signs, cash flows that change sign more than once can have as many IRRs as changes of sign. A large cost at the end, such as decommissioning, is the usual cause. The calculator then lists every IRR it finds and shows the one nearest the discount rate. In that case rely on NPV or MIRR instead.

Modified internal rate of return (MIRR)

MIRR = [ FV of the inflows at the reinvestment rate / PV of the outflows at the finance rate ]1/n − 1

The outflows are discounted to period 0 at the finance rate, and the inflows are compounded to period n at the reinvestment rate. MIRR always has exactly one value. It also avoids the IRR's built-in assumption that the money returned along the way earns the IRR itself. This is the same calculation as Excel's MIRR function.

Payback

Simple payback is the number of periods until the running total of the cash flows first turns from negative to zero or positive. Discounted payback is the same, with each cash flow first discounted at i. Within the period where the total turns positive, the calculator interpolates as if that period's cash flow arrived evenly through it. Payback ignores everything after the payback point, so use it as a screen for risk and liquidity, not to rank alternatives.

Profitability index and annual worth

Profitability index (PI) = PV of the inflows / PV of the outflows. A PI above 1 is the same as a positive NPV, and it shows the value returned per dollar invested.

Equivalent uniform annual worth (EUAW) = NPV × (A/P, i, n) = NPV × i / [1 − (1 + i)−n]

EUAW spreads the NPV evenly over the n periods, which is how alternatives with different lives are compared.

Example: is the high-efficiency rooftop unit worth it?

The present and future value calculator compares two 15-year rooftop units at a 6 % MARR. Unit 1 costs $40,000 installed, $9,000 a year to run, and needs a $6,000 compressor in year 8. Unit 2 costs $52,000, $7,500 a year to run, and has $4,000 salvage in year 15. Subtracting Unit 1 from Unit 2 gives the cash flows of the decision to pay more up front:

  • Period 0: −$12,000, the extra first cost
  • Periods 1 to 15: +$1,500 a year in lower running cost
  • Period 8: +$7,500, the $1,500 plus the $6,000 compressor Unit 2 avoids
  • Period 15: +$5,500, the $1,500 plus the $4,000 salvage

With (P/A, 6 %, 15) = 9.71225, (P/F, 6 %, 8) = 0.62741 and (P/F, 6 %, 15) = 0.41727:

NPV = −$12,000 + $1,500 × 9.71225 + $6,000 × 0.62741 + $4,000 × 0.41727 = $8,001.91

This is the difference in present worth found on that page, so Unit 2 is the better choice at 6 %. The other results:

  • IRR = 13.82 %, where NPV is zero. It is above the 6 % MARR, as the positive NPV says it must be.
  • MIRR at 6 % = 9.67 %. The $47,935.74 of inflows compounded to year 15 at 6 % against the $12,000 outlay gives (47,935.74 / 12,000)1/15 − 1.
  • Simple payback = 7.2 years. After 7 years the running total is −$12,000 + 7 × $1,500 = −$1,500, and year 8 returns $7,500, so 7 + 1,500 / 7,500 = 7.2.
  • Discounted payback = 7.77 years. The discounted running total is −$3,626.43 after year 7, and year 8 is worth $4,705.59 today.
  • PI = $20,001.91 / $12,000 = 1.667.
  • EUAW = $8,001.91 × 0.10296 = $823.90 a year.
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