Solve N, nominal annual interest, present value, payment or future value with signed cash flows, separate payment and compounding frequencies, and balance schedules.
01 / TIME VALUE OF MONEY
Cash received is positive (+); cash paid is negative (−). For a loan: positive PV and negative payments. For savings: negative deposits and positive FV.
Calculating PMT · periodic cash flow
Equal cash flow each payment period: received (+), paid (−). This variable is shown in the result; enter the other four below.
02 / VERIFIED TVM SOLUTION
Keep principal, N, payment timing, P/Y and C/Y fixed; solve PMT again using the same TVM engine. Differences mean alternative minus current, before fees.
| Period | Balance |
|---|---|
| 0 | $240,000.00 |
| 9 | $237,806.20 |
| 18 | $235,511.68 |
| 27 | $233,111.81 |
| 36 | $230,601.77 |
| 45 | $227,976.49 |
| 54 | $225,230.68 |
| 63 | $222,358.81 |
| 72 | $219,355.09 |
| 81 | $216,213.46 |
| 90 | $212,927.61 |
| 99 | $209,490.90 |
| 108 | $205,896.40 |
| 117 | $202,136.88 |
| 126 | $198,204.76 |
| 135 | $194,092.12 |
| 144 | $189,790.66 |
| 153 | $185,291.72 |
| 162 | $180,586.22 |
| 171 | $175,664.70 |
| 180 | $170,517.23 |
| 189 | $165,133.43 |
| 198 | $159,502.46 |
| 207 | $153,612.97 |
| 216 | $147,453.09 |
| 225 | $141,010.40 |
| 234 | $134,271.93 |
| 243 | $127,224.09 |
| 252 | $119,852.68 |
| 261 | $112,142.84 |
| 270 | $104,079.04 |
| 279 | $95,645.03 |
| 288 | $86,823.81 |
| 297 | $77,597.60 |
| 306 | $67,947.81 |
| 315 | $57,854.99 |
| 324 | $47,298.80 |
| 333 | $36,257.98 |
| 342 | $24,710.26 |
| 351 | $12,632.38 |
| 360 | $0.00 |
360 complete periods, monthly payments. Opening balance plus signed payment plus interest equals closing balance. Beginning payments occur before interest; ending payments occur afterward. Terminal FV is separate.
| Period | Opening | Payment | Interest | Closing |
|---|---|---|---|---|
| 1 | $240,000.00 | -$1,438.92 | $1,200.00 | $239,761.08 |
| 2 | $239,761.08 | -$1,438.92 | $1,198.81 | $239,520.96 |
| 3 | $239,520.96 | -$1,438.92 | $1,197.60 | $239,279.65 |
| 4 | $239,279.65 | -$1,438.92 | $1,196.40 | $239,037.12 |
| 5 | $239,037.12 | -$1,438.92 | $1,195.19 | $238,793.39 |
| 6 | $238,793.39 | -$1,438.92 | $1,193.97 | $238,548.43 |
| 7 | $238,548.43 | -$1,438.92 | $1,192.74 | $238,302.25 |
| 8 | $238,302.25 | -$1,438.92 | $1,191.51 | $238,054.84 |
| 9 | $238,054.84 | -$1,438.92 | $1,190.27 | $237,806.20 |
| 10 | $237,806.20 | -$1,438.92 | $1,189.03 | $237,556.31 |
| 11 | $237,556.31 | -$1,438.92 | $1,187.78 | $237,305.17 |
| 12 | $237,305.17 | -$1,438.92 | $1,186.53 | $237,052.77 |
| 13 | $237,052.77 | -$1,438.92 | $1,185.26 | $236,799.11 |
| 14 | $236,799.11 | -$1,438.92 | $1,184.00 | $236,544.19 |
| 15 | $236,544.19 | -$1,438.92 | $1,182.72 | $236,287.99 |
| 16 | $236,287.99 | -$1,438.92 | $1,181.44 | $236,030.51 |
| 17 | $236,030.51 | -$1,438.92 | $1,180.15 | $235,771.74 |
| 18 | $235,771.74 | -$1,438.92 | $1,178.86 | $235,511.68 |
| 19 | $235,511.68 | -$1,438.92 | $1,177.56 | $235,250.31 |
| 20 | $235,250.31 | -$1,438.92 | $1,176.25 | $234,987.64 |
| 21 | $234,987.64 | -$1,438.92 | $1,174.94 | $234,723.66 |
| 22 | $234,723.66 | -$1,438.92 | $1,173.62 | $234,458.36 |
| 23 | $234,458.36 | -$1,438.92 | $1,172.29 | $234,191.73 |
| 24 | $234,191.73 | -$1,438.92 | $1,170.96 | $233,923.77 |
r = (1 + I/Y ÷ (100 × C/Y))C/Y ÷ P/Y − 1
PV(1+r)N + PMT(1+r·begin) × ((1+r)N−1)/r + FV = 0
At r = 0: PV + N × PMT + FV = 0.
I/Y uses bracketed root isolation with extrema and a residual check. Multiple verified roots are rejected. Supported nominal rates are above −100% × C/Y through 10,000%, also bounded by |ln(1+r)| ≤20; N is above zero through 10,000. General cash flows do not automatically have a meaningful principal/interest split.
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Select exactly one TVM variable to solve and enter the other four. Set P/Y, C/Y and payment timing as needed, then review the verified solution, effective periodic rate and signed balance chart.
Solve any of the five TVM variables with consistent cash-flow signs and independent payment and compounding frequencies. Inspect loan repayment or investment growth, compare valid loan-rate scenarios, and verify balances in accessible schedules.
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PV is present cash flow, PMT is the periodic cash flow, FV is the terminal future cash flow, N is the number of payment periods, and I/Y is the nominal annual rate in percent. Exactly one is solved; the others are entered.
Cash received is positive and cash paid is negative from one consistent viewpoint. A borrower normally enters positive loan PV and negative repayments; a saver enters negative deposits and a positive final receipt. FV offsets the accumulated balance in the TVM equation.
The effective payment-period rate is (1 + nominal annual rate divided by C/Y) raised to C/Y divided by P/Y, minus one. The annual input percentage is converted to a decimal first. Annual compounding is not silently assumed when frequencies differ.
At zero rate, the equation is PV + N times PMT + FV = 0. Negative nominal rates are allowed while the compounding base stays positive and numerical limits are respected. Some cash-flow patterns have no solution or infinitely many solutions.
Yes. Nonconventional cash flows can have multiple rates. A bracketed exponential-polynomial root-isolation method checks extrema across the supported rate interval and verifies candidates against the stable TVM equation. Multiple or unverified solutions are not presented as a unique valid rate.
It is the mathematical duration of the annuity model, not an invented final partial payment. The calculator shows that interpretation explicitly. Detailed whole-period schedules are bounded to 600 periods; sampled chart values are available for larger supported cases.