Internal Rate of Return

The IRR is defined implicitly rather than explicitly. Setting the sum of each cash flow divided by one plus the rate raised to its period number equal to zero produces a polynomial of degree equal to the number of periods, and for most projects no closed-form solution exists. Practitioners therefore solve it iteratively: a rate is guessed, the resulting net present value is computed, and the guess is revised upward when NPV is positive and downward when NPV is negative until convergence.

Financial calculators automate this through the cash flow register, where the initial outlay is entered as a negative figure at time zero and subsequent flows in sequence. Spreadsheet functions do the same, though they require the analyst to supply an initial guess when the cash flow pattern is irregular. A candidate should recognise that the reported figure is a periodic rate: monthly cash flows produce a monthly IRR that must be annualised before comparison with an annual hurdle rate.

For a single conventional project evaluated in isolation, IRR and NPV always agree: if the IRR exceeds the required return, discounting at that required return produces a positive NPV. Conflict arises only when projects are mutually exclusive and differ in scale or in the timing of their cash flows. A small project returning a high percentage can create less absolute value than a large project returning a modest percentage, and a front-loaded project will show a higher IRR while a back-loaded project may generate more discounted value at low discount rates.

The crossover rate is the discount rate at which two projects have identical net present values, found by computing the IRR of the differential cash flow stream. Below the crossover rate the rankings differ; above it they coincide. Because NPV measures the absolute increase in firm value and is additive across projects, it is the theoretically correct tiebreaker whenever the two criteria disagree.

The IRR calculation implicitly assumes that interim cash flows are reinvested at the IRR itself, which becomes implausible when the computed rate is far above the firm's cost of capital. NPV, by contrast, implicitly assumes reinvestment at the required rate of return, a more defensible benchmark because it reflects the opportunity cost of capital actually available to the firm.

The modified internal rate of return addresses this directly. Interim inflows are compounded forward to the terminal date at the cost of capital to form a terminal value, outflows are discounted back to time zero at a financing rate, and the MIRR is the single rate that grows the present value of outflows into that terminal value. Because the resulting equation has one sign change, MIRR produces a unique solution and eliminates the multiple-root problem that afflicts non-conventional cash flow patterns.

The same mathematics recurs throughout the curriculum under different labels. A bond's yield to maturity is the IRR of its purchase price and promised coupon and principal payments. The money-weighted rate of return on a portfolio is the IRR of contributions, withdrawals and ending value, which is why it reflects the timing of investor cash flows rather than manager skill. Private capital performance, real estate underwriting and infrastructure concession bids all quote IRRs because the measure compresses an irregular cash flow profile into one comparable percentage.

A candidate should be able to compute an IRR from a cash flow stream, compare it to a stated hurdle rate, explain why the accept-reject signal matches NPV for independent projects, identify the source of a ranking conflict between mutually exclusive projects, and state clearly that NPV governs when the two diverge.