Financial calculator
Five variables, any four of which give the fifth: present value, future value, payment, rate and number of periods. That is the whole of the time value of money, and every loan, annuity and savings question on this site is one of those five being solved for. The rate has no closed form and is found by iteration.
Present value, payment, future value, number of periods, rate. Give any four and this returns the fifth — then puts all five back into the equation and shows you that they balance.
Cross-checked by test against the mortgage calculator · How we check
Payment
-$2,654.69
- Paid in or out over the term
- -$955,687.25
- Net cost
- $535,687.25
Check: the five values put back into the equation come to 0, so the answer balances rather than merely being printed.
The questions a payment calculator cannot answer
- “I can pay $450 a month for four years. What can I borrow?”
- $18,468.35 at 7.9%. This is solving for present value, and no calculator that fixes the payment as its output can do it.
- “$8,000 on a card at 19.99%, paying $250 a month. When is it gone?”
- 46.1 months — three years and ten months, costing $3,524.18 in interest. Drop the payment to $100 and there is no answer at all: the interest alone is more than that.
- “$420,000 at 6.5% over thirty years.”
- $2,654.69 a month, $535,686.85 of interest over the term — the same figure the mortgage calculator gives, which is checked by test in both directions.
Questions people actually ask
- Why are some numbers negative?
- Because the equation has to know which direction the money moves. Cash coming to you is positive and cash leaving you is negative, so a mortgage is a positive present value with negative payments: the bank hands you $420,000 and you hand back $2,654.69 a month. Make them all positive and nothing balances, which is why this page refuses those inputs instead of answering them.
- What rate do I enter for a 6.5% mortgage?
- 0.541667 — the annual rate divided by twelve, because the periods are months. The equation has no idea what a year is; it compounds once per period. Enter 6.5 with 360 monthly periods and you have described a loan at 6.5% a month, which is 114% a year.
- What does solving for the rate actually tell me?
- What you are being charged, as opposed to what you were told. Given the amount financed, the payment and the term, the rate falls out — and if it is above the quoted rate, the difference is fees. That is the same computation Regulation Z requires for a disclosed APR.
- What is the difference between payments at the start and the end?
- One period of interest. Rent, leases and insurance premiums are paid at the start of the period; loans are paid at the end. An annuity due is exactly the ordinary one discounted by a single period, so on a 0.5%-per-month loan the payment is 0.5% smaller.
- Why show a residual?
- Because a number printed by a calculator and a number that satisfies the equation are different claims. The residual is the five values fed back in: if it is not zero, the answer is wrong, and you can see that it is zero rather than take it on trust.
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