Loans

A loan is amortized with level or varying payments and you must find the principal paid, interest paid, or outstanding balance at a specific payment date.

Build the schedule row by row: each row's interest = balance × i, principal = payment - interest, new balance = old balance - principal. For a target payment, jump to that row using the prospective method (PV of remaining payments) rather than iterating all rows. Recognize that for a level-payment loan, principal portions grow geometrically.

Principal in payment t = P · v^{n - t + 1}; outstanding balance after t = P · a-angle-{n - t}.

After a partial repayment history, you must compute the outstanding balance at time t.

Prospective: present value of all remaining scheduled payments at the loan rate. Retrospective: accumulated original loan minus accumulated payments to date. Both should match for any well-defined amortization, so use both as a check.

OB_t (prospective) = P · a-angle-{n - t}; OB_t (retrospective) = L · (1 + i)^t - P · s-angle-t.

A loan pays only interest periodically while a separate sinking fund accumulates the principal repayment at a different rate.

Compute the periodic interest payment (loan principal × loan rate) plus the periodic sinking fund deposit (principal / s-angle-n at the sinking fund rate). Compare the total annual outlay to the level payment on an amortized loan at the same loan rate; the difference depends on the spread between loan and sinking fund rates.

Total periodic outlay = L · i + L / s-angle-n^{j}, where j is the sinking fund rate.