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Loan Payoff

Payment, total interest and the full amortization schedule of a fixed-rate loan (mortgage, car loan, student loan or a credit-card balance paid at a fixed amount), and what an extra payment each period does to the payoff date and the interest. The standard annuity formula gives the payment; the schedule is built period by period, so an extra payment, a lump sum or a balloon can be followed exactly. Charts show the balance against time with and without the extra payment and the split of each payment between interest and principal.

  • Payment from the annuity formula, total interest and total paid over the term

  • Extra payment per period or a one-time lump sum: new payoff time and the interest saved

  • Full amortization table (interest, principal, balance for every period) and a yearly summary

  • Charts of balance against time and of the interest-principal split

Loan Payoff

Unit System: /

Loan





Extra Payments




How the Payment and Schedule Are Computed | Method page
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Loan Payoff

How the Payment and Schedule Are Computed

The payment

A fixed-rate loan is an annuity: n equal payments M, one per period, whose present value at the periodic rate r equals the principal P0. Summing the geometric series gives M = P0 r / (1 − (1 + r)−n). The periodic rate is the nominal annual rate (the APR quoted by the lender) divided by the number of payments in a year; with monthly payments at 6 percent, r = 0.005. The total paid is M n and the total interest is M n − P0.

The schedule

Each period the interest due is the balance times r; the rest of the payment reduces the balance. The tool builds this schedule period by period rather than from the closed form, so that an extra amount added to every payment, or a lump sum at a chosen payment, is applied exactly: the extra goes to principal, the next period's interest is computed on the lower balance, and the loan ends when the balance reaches zero, with the last payment reduced to what remains.

Reading the results

Interest saved is the difference between the two schedules' total interest; years cut is the difference in their lengths. Because early payments are mostly interest, an extra amount paid early saves many times its own value in interest over a long loan. Biweekly payments at half the monthly amount make 26 half-payments a year, one extra monthly payment, which is the usual reason they shorten a mortgage.

Not included

Fees, escrow, points, variable rates, daily interest accrual conventions, prepayment penalties and taxes. A lender's schedule may differ by a few cents from rounding each payment to the cent.



References

Broverman, S. A. (2017). Mathematics of Investment and Credit, 7th ed. ACTEX Publications, Winsted, CT.

Kellison, S. G. (2009). The Theory of Interest, 3rd ed. McGraw-Hill Irwin, New York.

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