TVM 5 Keys Explained — N, I/Y, PV, PMT, FV Without the Confusion
Every financial calculator — BA II Plus, HP 12C, this site's engines — runs the same five-variable TVM machine. The math is one equation; what trips people up is the conventions around it: which keys get which numbers, which signs, and which period. This page is the map.
Quick answer: N = number of periods; I/Y = interest rate per period; PV = lump sum today; PMT = recurring payment each period; FV = lump sum at the end. Money you pay out is negative. The engine: PV(1+i)^n + PMT·[((1+i)^n−1)/i] + FV = 0 — store four keys, solve the fifth.
The five keys, one at a time
- N — the number of compounding periods, not necessarily years. 30 annual deposits is N = 30; a 5-year monthly loan is N = 60.
- I/Y — the interest rate per period. Annual problem, annual rate (8); monthly problem, monthly rate (6% a year → 0.5). On the BA II Plus you enter 8, not 0.08.
- PV — the value today of a single lump sum. Money you invest out is negative (you paid it); a loan you receive is positive.
- PMT — the level payment each period, end of period by default. Deposits are negative; payments you receive are positive. Zero for pure lump-sum problems.
- FV — the value at the end of the N periods, after the last payment. The account balance at maturity.
The sign rule (the #1 source of wrong answers)
The TVM equation balances flows through time, so each period must have money leaving you and money reaching you. Save $1,000 a year (PMT = −1,000) and FV comes back positive — the account pays you. Borrow $20,000 (PV = +20,000) and PMT comes back negative — you pay it. If your answer has the wrong sign or looks absurd, you almost certainly entered all five values with the same sign: no deal can have money only flowing one way.
A worked example you can retype anywhere
Invest $1,000 today at 8% for 3 years: PV = −1000, I/Y = 8, N = 3, PMT = 0, solve FV → +1,259.71. Same problem as a savings plan: deposit $1,000 at the end of each year for 30 years at 8% (PV = 0, PMT = −1000, N = 30, I/Y = 8) → FV = 113,283.21. Both run on this site's TVM calculator and match a BA II Plus to the cent.
Annual to monthly, in two moves
- Multiply periods: 5 years monthly → N = 60
- Divide the rate: 6% annual → I/Y = 0.5
- Check: $20,000 borrowed for 5 years at 6% (monthly) → PMT = −386.66
END vs BGN — the silent mode switch
END mode (ordinary annuity) pays at the end of each period; BGN (annuity due) at the start. Everything on this site, and the default of every handheld, is END. Rent and lease payments are the classic BGN case — paid up front. If a real BA II Plus shows 'BGN' in the corner, results will differ by a factor of (1+i) from this site's; switch it back before comparing.