Read the cash-flow timeline before comparing returns
Keep initial costs, numeric zero periods, actual dates and finance/reinvestment assumptions explicit.
| Period | Signed flow | Timing |
|---|---|---|
| 0 | -1000 | Immediate cost |
| 1 | 0 | A real zero-receipt period |
| 2 | 1210 | Terminal receipt |
A cash flow has a sign and a time
Negative values leave the project; positive values enter it. A cost paid immediately belongs to time zero. NPV discounts its first supplied value by one period, so add that immediate signed cost outside NPV. IRR uses the complete schedule, including the initial cost.
Do not remove a numeric zero from a periodic schedule. A zero receipt still occupies an interval. These rates are per declared period, which is not necessarily one year.
Use actual dates when row intervals differ
XNPV and XIRR use date-value pairs and actual day differences from the first date, divided by 365. Leap days change the elapsed days. Same-date receipts remain separate contributions; reordering later pairs together does not change the result.
Keep dates numeric and the first date earliest. Sorting only the dates destroys the mapping. A negative calculated yield can be valid when receipts do not recover the initial cost; hiding it with zero changes the answer.
State what happens to costs and receipts
MIRR discounts negative flows at the finance rate and compounds positive flows to the terminal period at the reinvestment rate. Divide terminal receipts by the positive present-cost magnitude, take the root over the number of intervals, then subtract one.
The two assumptions can differ. Neither a high IRR nor a supplied reinvestment rate establishes project quality, achievable borrowing terms or an investment recommendation. Different scale, duration, risk, taxes and fees require additional analysis.
Separate a unique teaching root from all possible IRRs
The first two tasks use one initial negative flow and nonnegative later receipts. This conventional pattern has one yield in their declared bracket. Multiple sign changes can introduce multiple roots or no useful root; a single solver output must not be treated as a general proof.
The MIRR task deliberately admits later costs and shows separate present-cost and terminal-receipt reports. Input validation keeps unsupported types and timing outside grading instead of silently normalizing them.
Check the idea
At a discount of 0.10 per period, what is the NPV at time zero of the example schedule?
Try the idea
References
Examples and explanations are original. Source links describe the related Excel features; native Excel workbook testing is separate from browser learning checks.