loan repayment

When you borrow money, the lender hands you a sum today and you promise a schedule of payments back. The central question of loan repayment is simple to state: what payments make the deal fair, so that the lender is fully repaid with the agreed interest and no more? The answer always comes from one principle — the amount borrowed today must equal the present value of all the payments you promise to make.

The most common arrangement is a level loan repaid by equal periodic payments, which is just an annuity-immediate from the lender's point of view. If you borrow L over n periods at rate i, each level payment equals L divided by a-angle-n. At any moment, the outstanding balance — what you would have to pay to clear the debt immediately — equals the present value of the payments still remaining (the prospective method), which also equals the original loan accumulated forward minus payments accumulated forward (the retrospective method); both give the same number. Interest in any period is i times the current balance, and the rest of the payment reduces principal.

Loan repayment mathematics underpins mortgages, consumer credit, corporate debt, and the bond market, and it is a staple of actuarial exams and practice. The same toolkit answers many practical questions: what is the payment, how much do I still owe, how much of this year's payment is interest for tax purposes, and what would it cost to pay off early. The honest caveat: advertised loan rates can hide fees and compounding conventions, so the true cost is found only by writing the full equation of value, not by trusting the headline rate.