Internal rate of return (IRR / TIR)

The internal rate of return — IRR, or TIR (taxa interna de retorno) — is the discount rate that makes an investment's NPV exactly zero. Intuitively, it is the project's own built-in rate of return.

The definition

IRR is the value of i that solves:

0 = CF0 + CF1/(1+i)^1 + CF2/(1+i)^2 + ... + CFn/(1+i)^n

There is no clean algebraic solution for most series — it is found by trial and error (or a financial calculator / spreadsheet). You guess a rate, compute NPV, and adjust.

The decision rule

Compare IRR to your required rate (the discount rate, also called the hurdle rate):

• IRR greater than hurdle — accept; the project beats your minimum.
• IRR less than hurdle — reject.

Worked example

The same project: -10,000, then 4,000, 4,000, 5,000. From the previous lesson we know:

• At 10%, NPV = +698.72 (positive)
• At 15%, NPV = -209.59 (negative)

So the IRR sits between 10% and 15% — that is where NPV passes through zero. Interpolating linearly between the two:

IRR is approximately 10% + 5% * 698.72 / (698.72 + 209.59)
IRR is approximately 10% + 5% * 0.7693 = 13.85%

Testing 13.8% confirms NPV is very close to zero, so the IRR is about 13.8%. If your hurdle rate is 10%, accept (13.8% > 10%); if it is 15%, reject (13.8% < 15%) — consistent with the NPV signs.

A caution

IRR is intuitive but has traps: a series whose cash flows change sign more than once can have multiple IRRs, and IRR can mislead when comparing projects of different size. When NPV and IRR disagree, trust NPV — it measures value created directly, in money.