Internal Rate of Return (IRR)

The internal rate of return is the discount rate at which the present value of an investment's inflows exactly equals the present value of its outflows, so that its net present value is zero. Office of Management and Budget Circular A-94, the federal guidance for benefit-cost analysis of government programs, states the definition in one line: "Internal Rate of Return. The discount rate that sets the net present value of the stream of net benefits equal to zero."

The definition is domain-general and worth reading that way. It appears in federal project appraisal, in real estate, in private-fund reporting, and on brokerage statements, and it means the same thing in all of them: the one constant annual rate that reconciles every dated cash flow with the amount invested. A bond's yield to maturity is the internal rate of return of that bond's cash flows, and the return figure a custodian computes from a client's own deposits and withdrawals is the internal rate of return of that account. Those are applications of the same arithmetic under different names, and each is a subject in its own right.

The metric is a solution, not a calculation, and that is the source of most of its behavior. Setting net present value to zero produces an equation in which the unknown rate appears once for every period, raised to a different power each time. That is a polynomial in the unknown rate, and past the fourth power a polynomial has no general formula for its roots at all, so for a stream of any realistic length there is nothing to rearrange. The answer is found by iteration instead: pick a rate, discount the flows, see whether the total comes out positive or negative, and move the rate in the direction that shrinks the gap. A spreadsheet's IRR function does exactly this, which is also why it asks for a starting guess and can fail to converge.

A stream that changes sign more than once can have several internal rates of return. Circular A-94 flags this in its own definition, adding that the figure "may have multiple values when the stream of net benefits alternates from negative to positive more than once." This is not a rounding artifact or a software bug. Each root genuinely sets net present value to zero, and nothing in the arithmetic nominates one of them as the answer. Any project with a large cost at the end (a well to be plugged, a building to be demolished, a lease with a terminal payment) has this shape, and so does a rental property bought, refinanced, and later sold at a loss.

The figure is silent about scale, which is why it is a poor decision rule on its own. Circular A-94 says so plainly at section 8.b(2): "While the internal rate of return does not generally provide an acceptable decision criterion, it does provide useful information, particularly when budgets are constrained or there is uncertainty about the appropriate discount rate." That last clause is the honest case for it. A percentage does not require you to name a discount rate first, so it is a way to describe an opportunity when reasonable people would pick different rates. What it cannot do is rank two opportunities of different sizes, because it reports a rate and the thing being compared is dollars.

Two reported figures are only comparable if the conventions behind them match, and the conventions genuinely differ. The choices that move the number include when each flow is dated (a payment treated as arriving at year end discounts differently from the same payment treated as arriving mid-year, and Circular A-94's appendix gives both), whether costs are stated before or after fees and taxes, and whether uninvested but committed money counts as invested yet. Federal regulation contains a deliberately different variant: 10 CFR 436.22 defines an "adjusted internal rate of return" for energy conservation measures, computed by compounding yearly savings forward to the end of the study period and comparing that terminal value with the present value of costs, which is a different calculation wearing a similar name. The practical reading is that an internal rate of return is a claim about a specific stream of dated cash flows, and a figure quoted without the stream behind it cannot be checked.

Lay out every cash flow with its date, treating money paid out as negative and money received as positive. Then find the rate that makes the discounted total come to zero. Because the equation cannot be rearranged, that means testing rates until the total crosses zero.

A hypothetical example of the multiple-root problem, chosen because the arithmetic is checkable by hand. A project costs $1,000 today, returns $2,600 at the end of year one, and then requires a $1,680 closing payment at the end of year two. Test 20 percent: the year-one inflow discounts to $2,600 ÷ 1.20 = $2,166.67, the year-two outflow discounts to $1,680 ÷ 1.44 = $1,166.67, and the total is −$1,000 + $2,166.67 − $1,166.67 = $0.00. Now test 40 percent: $2,600 ÷ 1.40 = $1,857.14, $1,680 ÷ 1.96 = $857.14, and the total is −$1,000 + $1,857.14 − $857.14 = $0.00 again. Both 20 percent and 40 percent are internal rates of return for the same project. Neither is wrong, and a spreadsheet will report whichever one its starting guess happens to find.

A hypothetical example of the scale problem. Deal A: pay $1,000 today, receive $1,300 in one year. Its internal rate of return is 30 percent, and at an 8 percent discount rate its net present value is $1,300 ÷ 1.08 − $1,000 = $1,203.70 − $1,000 = $203.70. Deal B: pay $50,000 today, receive $60,000 in one year. Its internal rate of return is 20 percent, and at the same 8 percent rate its net present value is $60,000 ÷ 1.08 − $50,000 = $55,555.56 − $50,000 = $5,555.56. The lower percentage is worth roughly twenty-seven times as many dollars.

Pros

• Compresses a whole stream of dated cash flows into one annual percentage, which is the form most people can reason about.
• Requires no discount-rate assumption to compute, so it is a way to describe an opportunity when reasonable people would choose different rates.
• Directly comparable to a required return: if the figure is below what you need from the risk taken, the deal does not clear the bar.
• Widely reported, so it is often the only summary number an offering actually discloses.

Cons

• Says nothing about size. A high percentage on a small outlay can be worth fewer dollars than a lower percentage on a large one.
• Can have more than one correct value when cash flows change sign more than once, and software reports only the one it finds first.
• Undefined for some streams: a project with no sign change has no rate that zeroes it.
• Sensitive to the dating and definition of the cash flows, so two figures are not comparable unless the conventions behind them are.
• Easy to present favorably, because the person choosing which flows to include and how to date them is usually the person selling the deal.