IRR Calculator

Formula and methodology

NPV(r) = Σ[t=0…n] C_t / (1+r)^t = 0
x = 1/(1+r) > 0; P(x) = Σ C_t x^t
Displayed IRR (%) = 100r per period

C_t is the signed flow at period t. Period 0 is now and is not discounted. Payments are negative, receipts positive. Only finite real rates greater than −100% are admissible here. Exact integer polynomial arithmetic, square-free reduction and Sturm sign variations count and isolate positive discount-factor roots. A GCD check detects repeated roots. Successful isolation covers every admissible root; no guessed preferred rate is selected.

Verified worked examples

−100 now and +110 one period later: −100 + 110/1.1 = 0, giving 10% per period. With monthly spacing it is a monthly rate. For −100, +230, −132, both 10% and 20% make NPV zero: −100 + 230/1.1 − 132/1.1² = 0; −100 + 230/1.2 − 132/1.2² = 0.

−100, +200, −100 gives −100(x−1)²: one distinct 0% IRR with a repeated root. −100, +50, −100 has no real IRR because the polynomial discriminant is negative. All-zero flows are indeterminate, not a 0% answer. Keeping a zero matters: −100, 0, +121 gives 10%, while −100, +121 gives 21% per period.

Inputs and interpretation

1. Choose one currency display and one equal interval. No FX conversion occurs.
2. Enter every signed cash flow in time order, including explicit zero amounts. Do not omit an elapsed period.
3. Add up to 25 entries. Removing the last period changes the timeline and clears the previous result.
4. Calculate in the local worker. Editing, clearing or cancelling invalidates active work; old responses cannot overwrite the new scenario.
5. Read the category first: roots, no admissible root, indeterminate, invalid or inconclusive. Multiple roots mean no unique IRR.

A large IRR does not establish low risk, causal benefits or the best use of capital. Changing every sign leaves the roots unchanged while reversing the perspective. Compare cash amounts, timing and relevant alternatives alongside any rate.

Assumptions and numerical limits

2–25 entries, each absolute magnitude at most 10^12, up to 8 fractional places and 32 characters. Zero and negative amounts are valid; missing values, grouping separators, scientific notation, dates, NaN and Infinity are rejected. All flows use the same equally spaced interval and currency basis. No day counts, annualization, reinvestment, inflation, automatic tax or fee calculations.

The exact solver has 200,000 counted arithmetic operations, an 8,192-bit monitored integer limit and 4,096 subdivisions. The worker has a five-second deadline. Exhaustion or loading failure produces inconclusive, never a claim of no IRR or a partial rate list. On success, refinement checks 10^−10 percentage-point interval width, 10^−16 relative discount-factor width and near-zero sign/relative resolution. Conversion uses 50 significant digits and half-even display rounding. Very close distinct rates can look equal at low display precision; the distinct-root count remains visible.

Financial amounts stay in browser memory, including worker messages. Solver assets load on first calculation; after loading, the worker can calculate offline. No main-thread or server fallback submits financial data. Independent CAS reference checks do not imply professional financial review or formal verification. Educational use only.

Frequently asked questions