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Loans and Amortization

Fixed monthly payments and amortization schedules.

Most car loans and mortgages are amortising loans: you pay the same amount every month, and each payment covers that month's interest first, with the rest reducing the principal. Early payments are mostly interest; later payments are mostly principal, because the balance the interest is charged on keeps falling.

The fixed payment comes from the present-value-of-annuity formula solved for the payment. An amortisation schedule lists each payment's split between interest and principal and the remaining balance. Extra principal payments shorten the loan and cut the total interest.

Key ideas

Payment: $M = P\frac{i(1 + i)^n}{(1 + i)^n - 1}$ , with $i$ the monthly rate and $n$ the number of payments.
Each month: interest $= i \times$ balance; principal paid $= M -$ interest.
Total interest $= nM - P$ .
Shorter terms mean higher payments but much less total interest.

Worked example

Find the monthly payment on a $20,000 loan at 6% annual interest over 5 years.

Monthly rate $i = 0.005$ ; payments $n = 60$ .
$(1.005)^{60} \approx 1.34885$ .
$M = 20{,}000 \cdot \frac{0.005 \times 1.34885}{0.34885} \approx 386.66$ .
Total interest: $60 \times 386.66 - 20{,}000 \approx 3199.60$ .

Answer: about $386.66 per month

Common mistakes

Using the annual rate in a monthly formula.
Thinking each payment reduces the principal by the full payment.
Ignoring the large total-interest cost of long terms.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.First-month interest on a $20,000 balance at 0.5% per month.
Show answer
$100
2.Monthly payment on $10,000 at 1% per month for 12 months, to two decimals.
Show answer
$888.49
3.Total interest on 60 payments of $386.66 for a $20,000 loan.
Show answer
$3,199.60

Key ideas

Worked example

Common mistakes

Practice questions

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