The five TVM values
- N: total number of periods, such as 360 months for a 30-year loan.
- I/Y: annual interest rate in percent. It is divided by payments per year for each period.
- PV: present value, the amount at the start (a loan amount or an initial deposit).
- PMT: the regular payment each period.
- FV: future value, the amount at the end (a savings goal or a balloon balance).
The formula
PV × (1 + i)^N + PMT × (1 + i·k) × [(1 + i)^N − 1] ÷ i + FV = 0
i = I/Y ÷ 100 ÷ payments per year, k = 1 in BEGIN mode, 0 in END mode
N, PV, PMT and FV can be solved directly from this equation. The interest rate has no closed formula, so the calculator searches for it numerically, as financial calculators do.
Worked examples
- Mortgage payment: N = 360, I/Y = 6.5, PV = 300,000, FV = 0 → PMT = −1,896.20 a month.
- Savings goal: N = 120, I/Y = 5, PV = 0, FV = 50,000 → PMT = −321.99 a month.
- Investment return: N = 10 (yearly), PV = −10,000, PMT = 0, FV = 20,000 → I/Y = 7.18%.
For specific goals, try the loan calculator, future value calculator or CAGR calculator.
Last reviewed October 2026.
Frequently asked questions
What is a TVM calculator?
A time value of money (TVM) calculator works with five values: number of periods (N), interest rate (I/Y), present value (PV), payment (PMT) and future value (FV). Enter any four and it solves for the fifth, just like the TVM keys on a financial calculator such as the BA II Plus or HP 12C.
Why do some values need to be negative?
TVM uses a cash-flow sign convention: money you pay out is negative and money you receive is positive. For a loan, the amount you borrow (PV) is positive and your payments are negative. For savings, your deposit (PV or PMT) is negative and the future value is positive.
What does payments per year do?
It sets how often payments and compounding happen. With 12 payments per year, N is the number of months and the annual rate is divided by 12 for each period. Use 1 for yearly problems.
What is the difference between END and BEGIN mode?
END mode (an ordinary annuity) assumes payments happen at the end of each period, which is normal for loans. BEGIN mode (an annuity due) assumes payments at the start of each period, which is common for rent and leases.
Why can't it solve for the rate sometimes?
A rate exists only if the cash flows change sign, for example you pay money in and get money back. If all values have the same sign, no interest rate can balance them. Check that the signs follow the convention.