The five keys
signs = cash-flow direction
solve for
compounding
payments
start from a scenario:
$
value today · + you have it, − you paid/promised it
$
value at the end · + you receive it, − you owe it
$
per compounding period · + received, − paid
n
compounding periods · e.g. 360 months with monthly
%
nominal annual · periodic rate = I/Y ÷ periods per year
updates live as you type
Solution
solving for · —
—
—
enter any four keys — the missing one is solved
FV + PV·(1+i)^N + PMT·(1+i·k)·[( (1+i)^N −1)/i] = 0
present value · PV—
future value · FV—
periodic payment · PMT—
periods · N—
annual rate · I/Y—
round-trip residual—
awaiting your numbers
Sign convention: money you receive or already have is positive; money you pay or deposit is negative. PV payments out, FV receipts in. Flip a sign if the solved value points the wrong way — a second root is often a mirror of the same deal.
how it’s calculated
| periodic rate · i | nominal I/Y ÷ periods/year — e.g. 6% monthly → 0.5%/month |
| growth factor | (1 + i)^N — lumpsum magnification over the term |
| annuity factor | [((1+i)^N − 1) ÷ i] × (1 + i·k) where k = 1 for begin-mode payments, 0 for end |
| master equation | FV + PV(1+i)^N + PMT·(1+i·k)·[((1+i)^N − 1)/i] = 0 |
| solve N | closed log form from the master equation |
| solve I/Y | closed form for lumpsum; bisection root-find for annuities, reported as nominal annual + effective annualized rate |
| effective annualized | (1 + i)^(periods/yr) − 1 — the true one-year yield including compounding |