Loans, compound growth, savings goals and rates of return on integers, with an amortisation schedule whose payments add up to the total and a balance that closes at exactly zero.
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You are considering a $20,000 loan at 5% nominal APR over 5 years and want the payment, the total, and figures that agree with each other.
Loan
$20,000 borrowed · 5% nominal, compounded monthly · 60 payments
Results
Per month $377.42 Final payment $377.74 Total paid $22,645.52 Interest $2,645.52 Effective annual rate 5.116190%
The exact level payment is 377.4247..., which nobody can hand over, so it is rounded to $377.42 and the last payment is $377.74 to clear the balance. The total is the sum of those sixty payments, so 59 x 377.42 + 377.74 = 22,645.52 exactly, and the page prints that identity as one of four it checks on its own output. Multiplying the unrounded payment by 60 instead would give $22,645.48.
Work out a loan payment, compound growth, a savings plan, a retirement pot, a rate of return or the effect of inflation, with every amount held as a whole number of the currency's smallest unit rather than as a floating-point number. The loan mode produces a real amortisation schedule: the interest is rounded each period the way a lender rounds it, the final payment absorbs the difference, and the balance closes at exactly zero.
Most loan calculators print a monthly payment rounded to the penny and then a total that is the UNROUNDED payment multiplied by the number of payments, so the two figures on the screen disagree. On a 300,000 loan at 6.5 percent over 30 years that gap is $1.47. Here the schedule is the calculation: 359 payments of the rounded amount plus a final payment that settles the balance, and the page prints four identities it has checked about its own output (the payments sum to the total, the interest plus the principal is the total, the principal repaid is the amount borrowed and the balance closes at zero). A zero interest rate is a real loan and produces a real answer, P divided by n, rather than a blank panel. Rates are always named: the nominal annual figure, the rate actually applied each period, and the effective annual rate (1 + j/m)^m - 1, which is the only one two products can be compared on. Eight currencies, each with its own ISO 4217 minor unit, so a yen loan is whole yen with no decimal point anywhere.
Two products quoted at different nominal rates and compounding frequencies are only comparable on the effective annual rate, which is printed next to the nominal one.
The schedule shows the interest and principal split for every payment, so a statement can be reconciled line by line.
Add an extra amount each period and the tool reports how many payments it removes and what it saves.
The savings-goal mode rounds the contribution up, runs the plan forwards on integers and shows where it really lands.
The internal rate of return mode finds every rate at which the cash flows cancel and warns when there is more than one.
Because nobody can pay 1896.204070478896 a month. The payment is rounded to a whole number of pennies, so the total is the sum of the payments actually made, and the last one is adjusted to clear the balance. Multiplying the unrounded payment by 360 gives a figure that is $1.47 away from what you would really hand over.
The nominal rate is the advertised annual figure. Divided by the number of periods it gives the rate applied each period, and compounding that back over a year gives the effective rate: (1 + j/m)^m - 1. At 6.5 percent compounded monthly the effective rate is 6.697185 percent, and that is the number two products have to be compared on.
Yes. An interest-free arrangement is a real loan and its payment is the amount borrowed divided by the number of payments. The annuity formula is 0/0 at a zero rate, so the tool uses the limit rather than refusing to answer.
It lists identities the page has computed about the numbers it just printed: that the payments add up to the total, that the interest plus the principal is the total, and that the balance closes at zero. Nothing external is needed to check those, so the page checks them and shows you the result.
Because the net present value is a polynomial in the discount rate, and Descartes' rule of signs says it can have as many positive roots as the cash flows have sign changes. A series that goes out, in, and out again can have two rates at which it cancels, and neither is more correct than the other.
Eight of them, each with the number of decimal places ISO 4217 gives it through CLDR. The yen has none, so a yen amount is a whole number with no decimal point and nothing to round away.
No. It is arithmetic. It knows nothing about tax, fees, or a market that does anything other than return the same percentage every period, and the retirement mode lists every assumption it is making on the page.
Every calculation runs locally in your browser. Your numbers and expressions are not transmitted or stored.