Learning goals
- Explain what a fixed installment payment covers (interest vs principal)
- Describe how amortization reduces balance over the term
- Compare scenarios using payment size and total interest, not payment alone
Fixed payment loans
Most consumer loans (auto loans, many personal loans, mortgages) use a fixed monthly payment. The lender calculates one payment so the balance reaches zero at the end of the term if you pay on schedule.
Each payment does two jobs: it pays interest owed for that period, and it reduces principal. Interest is charged on the remaining balance, so early payments are interest-heavy and later payments are principal-heavy.
Amortization is the schedule that lists every period’s interest, principal, and ending balance. Reading one row teaches more than memorizing the payment formula alone.
Standard installment payment
PMT = P × r(1 + r)^n / ((1 + r)^n − 1)
P is principal, r is the periodic rate (annual rate ÷ periods per year), and n is the total number of payments. For a 6% APR paid monthly, r = 0.06/12.
Interest in one period
interest = balance × r
Principal portion of the payment = payment − interest. New balance = old balance − principal portion. Repeat until the balance is zero.
Example 1 — First month interest vs principal
Given
Loan principal $10,000.00 6% APR, monthly payments, 3-year term. Approximate monthly payment ≈ $304.22. What happens in month 1?
Steps
- Monthly rate r = 0.06 / 12 = 0.005.
- Interest month 1 = 10,000 × 0.005 = $50.00.
- Principal month 1 ≈ 304.22 − 50.00 = $254.22.
- New balance ≈ 10,000 − 254.22 = $9,745.78.
Answer
Month 1 pays about $50.00 interest and $254.00 principal.
Tip: Nearly 1/6 of the first payment is interest even on a short 3-year loan.
Example 2 — Why early payments feel like “all interest”
Given
Same loan after many payments when the balance has fallen to $2,000.00.
Steps
- Interest on $2,000.00 at r = 0.005 → $10.00.
- If the payment is still ≈ $304.00 principal ≈ 304 − 10 = $294.00.
- Compare to month 1: interest dropped from $50.00 to $10.00 while principal rose sharply.
Answer
As balance falls, more of each fixed payment goes to principal.
Example 3 — Extra principal payment
Given
You pay an extra $500.00 toward principal in month 1 (after the regular payment).
Steps
- Extra principal reduces the balance immediately.
- Future interest = lower balance × r, so every later interest charge shrinks.
- The loan finishes earlier or future required payments cover more principal — total interest paid falls.
Answer
Extra principal saves interest because interest is always charged on what you still owe.
Tip: Confirm with your lender that extra amounts apply to principal, not prepaid interest or fees.
Comparing loans by payment only
Common mistake
Choosing the loan with the lowest monthly payment without checking term length or total interest.
Better approach
A longer term can lower the payment while increasing total interest. Compare APR, fees, term, and total interest paid.
Using annual rate as the monthly rate
Common mistake
Plugging 6% directly into a monthly formula as r = 0.06.
Better approach
For monthly payments, use r = APR / 12 (here 0.005). Periods and rate must match.
Check your understanding
1.If balance is $8,000.00 and monthly r = 0.004, what is next month’s interest before any payment?
Answer: 8,000 × 0.004 = $32.00.
2.A payment is $250.00 and interest for the period is $40.00. How much principal is paid?
Answer: 250 − 40 = $210.00.
3.Why do early amortization rows show higher interest?
Answer: Interest = rate × balance, and the balance is largest at the start.
Key takeaways
- Payment size depends on principal, periodic rate, and number of payments.
- Early schedule rows show higher interest and lower principal.
- Amortization tables list each period’s interest, principal, and balance.
- Extra principal payments reduce total interest and can shorten the term.