IRR and MIRR

The internal rate of return is the discount rate at which a project’s NPV is exactly zero — the rate at which the project just breaks even in present-value terms. It is a break-even test on the cost of money rather than on sales volume.

Because it is a rate, it comes with a decision rule that reads naturally to anyone running a business: borrow at 10%, earn 13%, take the project.

• For a conventional project — money out first, then a run of inflows — accept it if the IRR is above the cost of capital.
• Reject it if the IRR is below the cost of capital, because the project earns less than the money funding it costs.

That readability is the whole appeal of IRR, and it is why boards ask for it. A percentage can be compared against the bank’s lending rate without any further explanation, where an NPV of ZMW1,967 needs someone to say what it is 1,967 of.

There is no way to solve for IRR directly by hand on a project with several years of cash flows. Instead you calculate the NPV at two rates — one that leaves it positive and one that pushes it negative — and read off the point in between where the line crosses zero.

Take the same ZMW24,000 project from the last step. At 10% its NPV was ZMW1,967. Run it again at 14%.

The NPV crossed from positive to negative somewhere between the two rates, so that is where the IRR sits.

Watch the sign. Because b is negative, subtracting it adds: a − b = 1,967 − (−554) = 2,521. That denominator is the total distance the NPV travelled between the two rates, and the numerator is how much of that distance had been covered by the time it reached zero.

Against a 10% cost of capital the project is accepted, which agrees with the positive NPV — as it must, since both are reading the same cash flows.

IRR is the most widely quoted appraisal figure and the least reliable. Three problems, in rising order of seriousness.

The decision rule is not always clear cut

A project whose cash flows change direction more than once — an outflow, then inflows, then a large closure or restoration cost at the end — can have two IRRs, or more, or none at all. A mine with a rehabilitation obligation in its final year is the textbook case, and it is a common shape in Zambia. Told that a project has IRRs of 6% and 31%, the rule to accept when the IRR beats the cost of capital simply has no answer.

It is not the return on the project

IRR is routinely read as the return the project earns. It is not. The calculation assumes every cash flow the project throws off is immediately reinvested at the IRR itself for the rest of the project’s life. A project showing an IRR of 30% only earns 30% if the company really can find 30% homes for the cash as it arrives — and a company that could do that reliably would not need the project.

It does not compare projects

Because two projects can have different lives, different sizes and different reinvestment assumptions buried in their own IRRs, ranking them by IRR can put the wrong one first. A small project with a spectacular percentage can beat a large one that adds far more value, and the percentage gives you no way to see it.

MIRR keeps what is useful about IRR — a single percentage you can hold against the cost of borrowing — while removing the assumption that broke it. Instead of pretending cash is reinvested at the IRR, you state the rate at which the company can genuinely reinvest, and build the answer on that.

The method collapses the project into two numbers: what it costs today, and what it will be worth at the end.

• Take every outflow and discount it back to year 0 at the cost of capital. That is the investment phase, PVCF.
• Take every inflow and compound it forward to the final year at the reinvestment rate. That is the return phase, FVCF.
• Find the annual rate that grows PVCF into FVCF over the project’s life.
• Accept the project if that rate exceeds the cost of capital.

Two different rates appear in one calculation, and that is deliberate rather than sloppy. Money the company spends is funded at its cost of capital; money the company receives is put to work at whatever it can actually earn. Those are genuinely different rates, and IRR’s mistake was assuming they were the same.

Same ZMW24,000 project once more. The cost of capital is 10%, and the company can reinvest cash it receives at 12%. Compound each inflow forward to year 5 — a cash flow arriving in year 1 has four years left to grow, one arriving in year 5 has none.

The investment phase is simply the ZMW24,000, already at year 0. So the whole project reduces to a single question: what annual rate turns ZMW24,000 today into ZMW43,459 in five years?

12.6% is comfortably above the 10% cost of capital, so the project is accepted. Set the three answers side by side and the pattern is worth keeping.

MIRR comes in below IRR, and it will whenever the reinvestment rate is below the IRR. That gap is the size of IRR’s flattery. Where a project shows an IRR far above anything the company could realistically earn on spare cash, expect MIRR to be a good deal lower — and to be the number worth trusting.