Finance Calculator

This is the general five-variable time-value-of-money solver — the thing a financial calculator does when you press N, I/Y, PV, PMT and FV. Five quantities are locked together by one constraint, so you enter any four and pick which one to solve for. The constraint is PV × (1 + i)^n + PMT × (1 + i × type) × [(1 + i)^n − 1] ÷ i + FV = 0, where n is the number of periods, i is the periodic rate, PV is what the arrangement is worth today, PMT is the level payment made every period, FV is what settles the account at the end, and type is 1 for payments at the beginning of each period and 0 at the end. It is not asserted from nowhere: unrolling the recursion B_k = B_{k−1} × (1 + i) + PMT from a starting balance of PV and applying the geometric sum produces exactly that expression. At i = 0 the sum degenerates to n and the whole thing collapses to PV + PMT × n + FV = 0, which is handled as its own branch rather than by dividing by zero. Begin-versus-end timing is the setting people leave alone and shouldn’t. In the begin-of-period case every payment sits in the account one period longer, which multiplies the entire payment stream by (1 + i) — that is the whole of the (1 + i × type) factor. On a long monthly annuity it is worth roughly one extra month of interest on every payment you will ever make. Rent and lease payments are begin-of-period; almost every loan payment is end-of-period. The sign convention is what makes one equation cover saving, borrowing, leasing and annuities without special cases: money leaving you is negative, money arriving is positive. Depositing $10,000 today to receive $16,470.09 later is PV = −10000 and FV = +16470.09. Borrowing $5,000 and repaying $230 a month is PV = +5000 and PMT = −230. Get a sign wrong and the arithmetic is still perfect but the answer describes a different arrangement, which is the usual way a TVM calculator produces a confidently wrong number. Four of the five solve in closed form. The rate does not, so it is found by bisection, following Regulation Z, 12 CFR Part 1026 Appendix J (b)(9)(ii), which sanctions iteration and asks that all available decimals be carried. Appendix J publishes worked answers this implementation is tested against directly — PV = +1000, PMT = −33.61 over 36 periods gives 12.83% nominal, and PV = +5000, PMT = −230 over 24 periods gives 9.69%. The rate search covers 0% to 200% a year, a ceiling set to match the input cap so any answer you get back can be re-entered. The period count is the other awkward one: n appears only in an exponent, so it needs a logarithm, and the answer is deliberately left fractional — "how long until" rarely lands on a whole period. Two things here that most TVM tools skip. First, when no single rate fits, the calculator distinguishes three genuinely different failures instead of blaming your signs for all of them: the cash flows all point the same way and nothing can balance them; the implied rate is outside the supported range; or — by Descartes’ rule of signs, which bounds this problem at two positive roots — there really are two rates that balance your flows and reporting one would be a lie. Second, all five solved values are substituted back into the constraint and the leftover residual is printed, so you can see for yourself that the answer closes. Rates are quoted annually and divided by your chosen periods per year (1, 2, 4, 12, 52 or 365), matching Appendix J (b)(1)’s nominal convention, and a solved rate is multiplied back the same way so a solve round-trips exactly. For loan-shaped problems specifically, the amortization calculator shows the payment-by-payment split, and the annuity calculator handles payout structures with more built-in assumptions than this one makes.