YieldraWork through a two-root example
Consider equally spaced annual cash flows of −100 now, +230 after one year and −132 after two years. The NPV equation is −100 + 230/(1+r) − 132/(1+r)^2 = 0.
Both r = 10% and r = 20% satisfy that equation. At 10%, the discounted terms are −100, about +209.09 and about −109.09. At 20%, they are −100, about +191.67 and about −91.67. Each set sums to approximately zero.
Read the profile instead of selecting your favourite rate
A numerical solver may converge on a root near its starting guess. Finding 10% does not disprove 20%. A profile plotting NPV across discount rates makes the two crossings visible in this constructed example.
Multiple sign changes permit complications, but do not guarantee a specific number of real, economically relevant roots. Some cash-flow patterns have no useful IRR at all. Inspect the calculation's domain as well as its output.
Use a discount rate for an explicit NPV question
If an exercise supplies a required discount rate, calculate NPV at that rate and state the assumption. MIRR is another measure with explicit financing and reinvestment assumptions; it should not be introduced as an assumption-free replacement.
Yieldra's live listing describes an NPV profile, multiple-IRR warnings and cash-flow worksheets. These are Pro features, alongside MIRR results. Keep the original flows and timing with any exported answer so that someone else can reproduce the calculation.
- Check every cash-flow sign.
- Keep equal-period IRR separate from date-based calculations.
- Do not treat a solver's first root as a complete conclusion.