Maps for Math

How do I know if an investment is worth it, and what is IRR?

Discount each year's cash by (1 + r) per year and add it up. That's net present value, and it's a polynomial in 1/(1 + r). If it's positive at the return you need, the investment beats your alternative. The rate that makes it exactly zero, the polynomial's root, is the internal rate of return.

−$40know$8kyr 1$8kyr 2$8kyr 3$8kyr 4$20kyr 5IRR 8.09%At your 10%: NPV −$2.22k
Net present value at your rate
−$2,222.65
Internal rate of return (IRR)
8.09%
Rates where NPV = 0
1
Year the money comes back
5

Challenge: With the cleanup-bill job, find a required return where it actually pays off.

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What you could earn elsewhere with similar risk. A savings account might be 4%, the stock market’s long-run average around 7 to 10%.

Play

Load the food truck, then slide your required return up until the net present value hits zero.

Challenge: With the cleanup-bill job, find a required return where it actually pays off. The box under the picture turns green when you get it.

Stuck? Pick one of the examples from the “Try an example” menu, or press “New example.”

Understand

NPV(r)=∑t=0TCt(1+r)t=∑t=0TCt xt,x=11+r\text{NPV}(r) = \sum_{t=0}^{T} \frac{C_t}{(1 + r)^t} = \sum_{t=0}^{T} C_t\,x^t,\quad x = \frac{1}{1 + r}

A dollar next year is worth less than a dollar today, because today's dollar could be earning rr. So divide each year's cash by (1+r)t(1 + r)^t and add:

NPV(r)=∑t=0TCt(1+r)t\text{NPV}(r) = \sum_{t=0}^{T} \frac{C_t}{(1 + r)^t}

Call x=1/(1+r)x = 1/(1 + r) and it's just a polynomial, C0+C1x+C2x2+…C_0 + C_1x + C_2x^2 + \dots The internal rate of return is where the curve crosses zero, its root. A polynomial can have several roots, so an investment can have several IRRs.

Use

Every input has a unit menu, so you can type values in the units you already have. Results follow your units.

Show the work

  1. Money later is worth less today: divide year t by (1 + r)^t\text{NPV} = -40{,}000 + \frac{8000}{(1 + 0.1)^{1}} + \frac{8000}{(1 + 0.1)^{2}} + \frac{8000}{(1 + 0.1)^{3}} + \frac{8000}{(1 + 0.1)^{4}} + \frac{20{,}000}{(1 + 0.1)^{5}}
  2. Add it up\text{NPV} = -2223
  3. Write x = 1/(1 + r), and NPV is a polynomial in x\text{NPV} = -40{,}000 +8000x +8000x^{2} +8000x^{3} +8000x^{4} +20{,}000x^{5},\quad x = 0.9091
  4. Sign changes in the cash flows (Descartes’ rule: at most this many positive-rate roots)1
  5. IRR: the roots, where NPV = 0r = 8.089\%

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Set what goes in and comes out each year, and the return you need.

  • Taxes, inflation, and risk are left out, and future amounts are estimates.

For learning and estimation. Verify with applicable codes, standards, and a qualified professional before using in design, construction, or safety-critical work.

Cheat card

NPV(r)=∑t=0TCt(1+r)t\text{NPV}(r) = \sum_{t=0}^{T} \frac{C_t}{(1 + r)^t}
NPV=∑Ct xt,x=11+r\text{NPV} = \sum C_t\,x^t,\quad x = \frac{1}{1 + r}
IRR: NPV(r)=0\text{IRR: NPV}(r) = 0
SymbolMeaningUnit
CtC_tcash in (+) or out (−) in year t$
rrthe return you need each year%
xxdiscount factor, 1/(1 + r)
  • Positive NPV at your rate means it beats your next-best option.
  • If the IRR is above your required return, a normal investment is worth doing.
  • Cash that changes direction more than once can give more than one IRR, so check the NPV curve.

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Where it’s used

  • Finance & Business
    Analysts rank projects, buildings, and companies by NPV and IRR.
  • Finance & Business
    Mining and drilling projects with cleanup costs are the classic case of multiple IRRs.
  • Money & Shopping
    Compare a side business, a rental, or solar panels against simply saving the money.

Questions people ask

What is IRR?

The yearly rate of return that makes an investment exactly break even in today's money. It lets you compare investments of different sizes and timings.

What's the difference between NPV and IRR?

NPV is a dollar amount at the return you choose. IRR is the rate where NPV is zero. NPV tells you how much you gain, and IRR tells you how fast.

How can an investment have two IRRs?

NPV is a polynomial, and a polynomial can cross zero more than once. That happens when the cash changes direction twice, like a big cleanup bill at the end.

Sample PLY-0002 · rev 1 · model PLY-M02 · Level 4 · advanced algebra · updated 2026-09-29