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Annuity Due Formulas
These are the main formulas that are needed to work with regular annuity cash flows (Definition/Tutorial). Please note that these formulas work only on a payment date, not between payment dates. This is the same restriction used (but not stated) in financial calculators and spreadsheet functions.
I use MathJax to display these formulas. You can click the equations to show a zoomed version in a pop-up, or right-click on the formulas for additional options (e.g., to copy the equation as $\TeX$ or to choose the zoom settings).
| To Solve For: | Formula |
|---|---|
| Future Value | $$F{V_{AD}} = Pmt\left[ {\frac{{{{\left( {1 + i} \right)}^N} – 1}}{i}} \right]\left( {1 + i} \right)$$ |
| Present Value | $$P{V_{AD}} = Pmt\left[ {\frac{{1 – \frac{1}{{{{\left( {1 + i} \right)}^{\left( {N – 1} \right)}}}}}}{i}} \right] + Pmt$$ |
| Periodic Payment When PV is Known | $$Pm{t_{AD}} = \frac{{P{V_{AD}}}}{{\left[ {\frac{{1 – \frac{1}{{{{\left( {1 + i} \right)}^{\left( {N – 1} \right)}}}}}}{i} + 1} \right]}}$$ |
| Periodic Payment When FV is Known | $$Pm{t_{AD}} = \frac{{F{V_{AD}}}}{{\left[ {\frac{{{{\left( {1 + i} \right)}^N} – 1}}{i}} \right]\left( {1 + i} \right)}}$$ |
| Number of Periods When PV is Known | $${N_{AD}} = \frac{{ – \ln \left( {1 + i\left( {1 – \frac{{P{V_{AD}}}}{{Pm{t_{AD}}}}} \right)} \right)}}{{\ln \left( {1 + i} \right)}} + 1$$ |
| Number of Periods When FV is Known | $${N_{AD}} = \frac{{\ln \left( {1 + \frac{{F{V_{AD}}}}{{Pmt\left( {1 + i} \right)}}i} \right)}}{{\ln \left( {1 + i} \right)}}$$ |
| Discount Rate | Can only be found through trial and error (e.g., the bisection method or Newton’s method) |
| Variable | Definition |
|---|---|
| $$FV$$ | Future Value |
| $$PV$$ | Present Value |
| $$i$$ | Discount Rate per Period |
| $$N$$ | Number of Periods |
| $$Pmt$$ | Annuity Payment per Period |
| $$\ln{()}$$ | Natural Logarithm ($\log_e$) |