Internal rate of return

The internal rate of return (IRR) of a project is the interest rate at which the project’s present worth is exactly zero. Equivalently, it’s the rate at which the project’s discounted benefits exactly equal its discounted costs over the project’s life.

$PW={\sum }_{t=0}^T\frac{R_t-D_t}{(1+i^{\ast }{)}^t}=0$

where $R_t$ are receipts (positive cash flow), $D_t$ are disbursements (negative cash flow), and $i^{\ast }=$ IRR is the root of the equation.

The IRR is “internal” because it depends only on the project’s own cash flows — it doesn’t reference any external rate. It captures the project’s intrinsic earning power. If the project clears the firm’s MARR, it’s economically attractive:

$\text{Accept if }\text{IRR}>\text{MARR},\text{ else reject.}$

Calculation. In all but trivial cases there’s no closed form for IRR — it’s the root of a polynomial in $(1+i^{\ast })$, generally requiring numerical solution. In practice you use Excel’s IRR() function or Newton’s method or trial-and-error. For simple two-flow projects (one cost, one revenue) the IRR is direct: if $-P$ now becomes $F$ at year $N$, then $\text{IRR}=(F/P{)}^{1/N}-1$.

Graphical view. Plot PW as a function of the discount rate. PW is high at low rates (future cash flows dominate) and low (or negative) at high rates. The IRR is where the curve crosses zero. For “well-behaved” cash flows (one big negative followed by a sequence of positives), the curve crosses zero exactly once and the IRR is unambiguous.

The multiple-IRR problem

If cash flows change sign more than once (negative → positive → negative, for example), Descartes’ rule of signs gives an upper bound on the number of positive real roots equal to the number of sign changes — and the actual count differs from that bound by an even integer. Two sign changes therefore mean 0 or 2 positive real IRRs; three sign changes mean 1 or 3. Some of the roots may be mathematical artifacts with no economic meaning, and some patterns produce no real IRR at all.

Common case: a mining project — large positive cash flows in operating years but a large negative cash flow at the end for environmental remediation. The PW equation has two sign changes, hence 0 or 2 IRRs.

When this happens, IRR by itself is unreliable. Use ERR instead, which fixes the pathology by assuming reinvestment of interim flows at the MARR.

Standard rule of thumb: for IRR to be well-defined, cash flows should change sign at most once.

Incremental analysis for mutually exclusive projects

The IRR has a subtle pitfall for ME alternatives: the project with the highest IRR is not necessarily the right pick. IRR rewards small high-return projects; a larger project with slightly lower IRR can deliver more total value.

The fix is incremental analysis:

1. Sort the ME projects by first cost, lowest to highest.
2. The lowest-cost project is the current best.
3. Challenge it with the next-costlier project. Compute the IRR of the incremental cash flows (challenger minus current best). This is the rate of return on the extra money spent.
4. If the incremental IRR exceeds MARR, the challenger wins (the extra money is well spent). Update the current best.
5. If not, the incremental investment doesn’t clear MARR; keep the current best.
6. Repeat from step 3 with the next project on the list.

The final current best is the optimal ME pick. This is equivalent to picking the project with the highest PW at MARR — but framed in IRR language for shops that prefer rates of return over dollar amounts.

For IRR’s companion that handles multiple-IRR projects, see External rate of return. For the PW method that doesn’t have this pitfall, see Present worth method.