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Write the equally spaced equation
The displayed domain keeps the per-interval gross factor positive; every t refers to the same interval length.
| Input | Meaning |
|---|---|
| CF_0 | Net immediate amount, not discounted |
| CF_t | Net flow at interval-end t |
| r | A rate for one modeled interval |
| T | Final represented interval, including zero-flow gaps |
Inspect a three-date case
For −100, 230 and −132, let a = 1 + r. Multiplying by a² gives −100a² + 230a − 132. It factors as −100(a − 1.1)(a − 1.2), so two admissible gross factors and two interval rates satisfy the equation.
Do not infer roots solely from a label
The number of sign changes can provide information about possible roots, but does not certify a unique root in a chosen range. Numerical routines have supported ranges and iteration behavior. Check a reported candidate by substituting it into the original NPV equation.
Report the timing and root together
A 10% quarterly root and a 10% annual root describe different intervals. Converting a root to another quotation is an additional step with explicit assumptions. Do not omit the interval unit when reporting an IRR.