Internal Rate of Return (Newton)
Solve the discount rate that zeroes an even-interval cash-flow series — Newton-Raphson on the NPV equation.
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The engineering
IRR is the discount rate that makes the net present value of a cash-flow series exactly zero — the break-even return where money grows fast enough to justify the up-front outlay. Enter the flows in time order starting at t=0 (typically the negative investment), one entry per equal period; the card runs Newton-Raphson on NPV(i) starting from your guess.
Sanity check: IRR only makes sense when the series changes sign at least once (you spend, then you earn). A series with multiple sign changes can have multiple IRRs — Newton lands on whichever root is nearest the guess, so nudge the initial rate if the answer looks wrong. Compare IRR against your MARR (minimum acceptable rate of return): above it the project pays, below it it doesn't.
Where this math comes from
Discounting future money to a present worth is old — Leibniz sketched the idea in the 1680s and actuaries used it for centuries — but IRR as a project-ranking number belongs to the 20th-century capital-budgeting world. Kenneth Boulding gave a clean statement of the internal rate in 1935, and Joel Dean's 1951 book Capital Budgeting pushed it into corporate practice as the yardstick for whether an investment beats the cost of capital.
The Newton-Raphson root-finding underneath is older still, from Isaac Newton's late-1600s method and Joseph Raphson's 1690 published form. Blank and Tarquin's Engineering Economy is where most working engineers first meet IRR wired to a trial-and-error or Newton solve — the exact procedure this card automates.
- 1690Joseph RaphsonPublishes the iterative root-finding method later paired with Newton's name.
- 1935Kenneth BouldingGives a formal statement of the internal rate of return.
- 1951Joel DeanCapital Budgeting popularizes IRR as a corporate investment criterion.
- 2018Blank & TarquinEngineering Economy (8th ed.) codifies the IRR/NPV workflow used here.
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