About the Financial Calculator (TVM)
This financial calculator solves time value of money (TVM) problems. The same five variables cover mortgage and car loan payments, how much a savings plan grows to, how long it takes to reach a goal, how much you can borrow, and the true interest rate hidden in a lease or installment offer. Payments per year (P/Y) and compounding periods per year (C/Y) can be set separately, and payments can be made at the end (ordinary annuity) or the beginning (annuity due) of each period.
It is handy for finance, accounting and CFA/CFP coursework when your calculator isn't at hand, or for double-checking answers.
How to use
- Choose the value to solve for (N, I/Y, PV, PMT or FV). That box becomes the result.
- Fill in the other four. Use the sign convention: money out is negative, money in is positive.
- Set payments per year (12 for monthly), compounding per year and END or BEGIN.
- Read the result, total payments, total interest, effective annual rate and the year-by-year balance table.
The TVM equation and the sign convention
With the rate per period i = (1 + I/Y ÷ 100 ÷ C/Y)C/Y ÷ P/Y − 1 and t = 1 in BEGIN mode (0 in END mode), the five values always satisfy:
PV × (1+i)N + PMT × (1 + i·t) × ((1+i)N − 1) ÷ i + FV = 0
Because PV, PMT and FV sit in the same equation, cash you receive must be positive and cash you pay negative. If all three have the same sign the equation can never equal zero – that is why a financial calculator shows an error when solving for I/Y or N. There is no closed formula for the rate, so I/Y is found numerically (Newton's method with a bisection fallback).
Worked examples
| Problem | Inputs | Answer |
|---|---|---|
| 30-year mortgage payment | N 360, I/Y 6.5, PV 300,000, FV 0 | PMT −$1,896.20 |
| $200,000 at 6% for 30 years | N 360, I/Y 6, PV 200,000, FV 0 | PMT −$1,199.10 |
| Save $500 a month for 10 years at 5% | N 120, I/Y 5, PV 0, PMT −500 | FV $77,641.14 |
| Same, deposits at the start of each month | … + BEGIN | FV $77,964.64 |
| Car loan rate: $20,000, 60 × $386.66 | N 60, PV 20,000, PMT −386.66, FV 0 | I/Y 6.00% |
When P/Y and C/Y differ
Canadian mortgages, for example, are paid monthly (P/Y 12) but compounded semiannually (C/Y 2), so the rate per period is (1 + rate/2)2/12 − 1. Most US loans and savings accounts quoted with an APR use the same frequency for both, so leave C/Y at “Same as P/Y”. The effective annual rate (EAR, or APY for deposits) is (1+i)P/Y − 1.
FAQ
Why do I get “no interest rate fits these cash flows”?
PV, PMT and FV probably all have the same sign. For a loan, the amount borrowed (PV) is positive and the payments (PMT) are negative. Without at least one sign change there is no rate that balances the equation – the same reason a BA II Plus shows Error 5.
How do END and BEGIN differ?
END (ordinary annuity) means payments at the end of each period, as with most loans and savings plans. BEGIN (annuity due) means payments at the start, as with rent, leases and insurance premiums. BEGIN earns one extra period of interest on every payment.
Why is my lender's payment slightly different?
Lenders may count actual days, round each month to the cent or add fees and escrow. This calculator uses the standard TVM formula without intermediate rounding, which matches financial calculators.
Can N be a fraction?
Yes. A fractional N means the final payment is a partial one. The yearly table rounds N up to show the last period.