### Determining i or n for Known Cash Flow Values

**Single Amounts — P and F Only**

**Hand or Calculator Solution**Set up the equivalence relation and (1) solve for the variable

using the factor formula, or (2) find the factor value and interpolate in the tables. Spreadsheet Solution Use the IRR or RATE function to find i or the NPER function to find n . (See below and Appendix A for details.)

**Uniform Series — A Series**

**Hand or Calculator Solution**Set up the equivalence relation using the appropriate factor

( P/A , A/P , F/A , or A/F ), and use the second method mentioned above. Spreadsheet Solution Use the IRR or RATE function to find i or the NPER function to find n .

**Mixed A Series, Gradients, and/or Isolated Values**

**Hand or Calculator Solution**Set up the equivalence relation and use (1) trial and error or (2) the calculator functions.

**Spreadsheet Solution**Use the IRR or RATE function to find i or the NPER function to find n . (This is the recommended approach.)

Besides the PV, FV, and NPV functions, other spreadsheet functions useful in determining i are IRR (internal rate of return) and RATE, and NPER (number of periods) to find n . The formats are shown here and the inside front cover with a detailed explanation in Appendix A. In all three of these functions, at least one cash flow entry must have a sign opposite that of others in order to find a solution.

To use IRR to find i , enter all cash flows into contiguous cells, including zero values.

The single-cell RATE function finds i when an A series and single P and/or F values are involved.

NPER is a single-cell function to find n for single P and F values, or with an A series.

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