What is an amortized loan (simple definition)
– An amortized loan is a debt that is repaid by a series of scheduled payments that each cover some interest and some principal (the original amount borrowed). Over the life of the loan the share of each payment that goes to interest falls while the share that reduces principal rises.
Key terms (defined)
– Principal: the remaining unpaid balance of the loan.
– Interest rate: the cost of borrowing, usually quoted as an annual percentage rate (APR).
– Periodic rate: the interest rate for one payment period (e.g., monthly rate = APR/12 for monthly payments).
– Amortization schedule: a table showing, for each payment period, the interest paid, principal paid, and the remaining balance.
– Prepayment: any payment that reduces principal in addition to the required scheduled payment.
– Balloon loan: a loan with a large final payment (see comparison below).
– Revolving credit: a borrowing arrangement (e.g., credit card) where you can reuse credit as you repay it.
How amortization works (mechanics)
1. Convert the stated annual interest rate into the periodic rate you’ll use for calculations (for monthly payments divide by 12).
2. Find the fixed payment amount that will fully repay the loan over the agreed number of periods.
3. For each period:
– Interest due = current balance × periodic rate.
– Principal paid = fixed payment − interest due.
– New balance = current balance − principal paid.
4. Repeat until the balance is zero.
Formula for the fixed payment (for fixed-rate, fully amortizing loans)
– Monthly payment = P × r / (1 − (1 + r)^−N)
– P = initial principal
– r = periodic interest rate (APR/12 for monthly)
– N = total number of payments (years × 12 for monthly)
Assumptions: fixed interest rate, fixed payment schedule, no fees, payments on schedule.
Short, worked numeric example
– Loan: $165,000 mortgage
– Term: 30 years → N = 30 × 12 = 360 monthly payments
– APR: 4.5% → monthly rate r = 0.045/12 = 0.00375
Step 1 — compute the monthly payment:
– Monthly payment = 165,000 × 0.00375 / (1 − (1 + 0.00375)^−360)
– Numerically this gives ≈ $836.60 per month (rounded).
Step 2 — first payment breakdown:
– Interest in month 1 = 165,000 × 0.00375 = $618.75
– Principal in month 1 = 836.60 − 618.75 = $217.85
– New balance after payment 1 = 165,000 − 217.85 = $164,782.15
Notes from the example
– Early payments are heavily weighted to interest. Here roughly 74% of the first payment is interest.
– As principal declines, interest each month falls, so future payments put more toward principal.
Can you pay off an amortized loan early?
– Yes. Extra principal payments reduce the outstanding balance immediately, shorten the remaining term, and lower total interest paid.
– Before making extra payments, check the loan contract for any prepayment penalties or lender rules.
– Making extra principal payments does not automatically change the fixed monthly payment unless you refinance or otherwise modify the loan terms.
How to see how much of a payment is interest
– Ask the lender for an amortization schedule, which should list interest and principal by period.
– Or compute interest for the current period using: interest = current balance × periodic rate.
Do you pay more interest at the beginning or at
the end of the loan? — You pay more interest at the beginning.
Why: each periodic interest charge equals the outstanding balance times the periodic rate. Early in the schedule the balance is largest, so the interest portion of each fixed payment is largest. Over time the principal portion of each payment grows, because each payment reduces the outstanding balance and thus the subsequent interest charge.
Key formulas (definitions first)
– Periodic rate (r): annual interest rate divided by periods per year (e.g., 4% / 12 = 0.0033333).
– Number of periods (n): years × periods per year (e.g., 30 × 12 = 360).
– Fixed periodic payment (A) for a fully amortizing loan:
A = r × PV / (1 − (1 + r)^−n)
where PV is the loan principal (present value).
– Interest in period t:
Interest_t = Balance_{t−1} × r
– Principal in period t:
Principal_t = A − Interest_t
– New balance after payment:
Balance_t = Balance_{t−1} − Principal_t
Worked numeric example
Assume a $200,000 mortgage, 30-year term, 4.0%