Borrowing and paying back
A loan lets you borrow money now and repay it over time with interest. Most consumer loans (car loans, student loans, mortgages) use amortization: equal regular payments that each cover the interest due plus some principal.
The payment formula
For a loan of PV with interest rate i per period over n payments:
PMT = PV × i / [1 − (1 + i)⁻ⁿ]
Example: $20,000 car loan at 6% per year compounded monthly, over 5 years (60 months).
- i = 0.005, n = 60
- PMT = 20,000 × 0.005 / [1 − 1.005⁻⁶⁰] ≈ $386.66/month
Financial calculators, spreadsheets (=PMT) and online calculators do this quickly.
Amortization schedule
An amortization schedule shows how each payment is split:
| Month | Payment | Interest | Principal | Balance |
|---|---|---|---|---|
| 0 | $20,000.00 | |||
| 1 | $386.66 | $100.00 | $286.66 | $19,713.34 |
| 2 | $386.66 | $98.57 | $288.09 | $19,425.25 |
| 3 | $386.66 | $97.13 | $289.53 | $19,135.71 |
Interest = balance × i (20,000 × 0.005 = $100). Principal = payment − interest.
Early payments are mostly interest; later payments are mostly principal.
Total interest
Total interest = (payment × number of payments) − amount borrowed
386.66 × 60 − 20,000 ≈ $3,199.60
Effect of the term
The term (length) changes the payment and total interest.
| Term | Monthly payment | Total interest |
|---|---|---|
| 3 years | ≈ $608.44 | ≈ $1,903.79 |
| 5 years | ≈ $386.66 | ≈ $3,199.60 |
| 7 years | ≈ $292.17 | ≈ $4,542.37 |
Longer terms mean smaller payments but much more interest.
Effect of the rate
On the same $20,000 over 5 years: at 4%, payment ≈ $368.33; at 9%, ≈ $415.17. A few percentage points add up to hundreds or thousands of dollars.
Paying extra
Extra payments go straight to principal, shortening the loan and reducing total interest.
Worked examples
Example 1: Balance $10,000, monthly rate 0.5%. Interest for this month? $50.
Example 2: Payment $250, interest $50. Principal portion? $200.
Example 3: 48 payments of $450 on a $18,000 loan. Total interest? 21,600 − 18,000 = $3,600.
Common mistakes
- Using the annual rate in monthly calculations.
- Thinking each payment reduces the balance by the full payment.
- Choosing the longest term without considering total interest.
Try it yourself
- Balance $15,000, annual rate 7.2% compounded monthly. First month’s interest?
- Payment $300; interest $90. New balance from $15,000?
- 36 payments of $500 on a $16,000 loan. Total interest?
- Why are early payments mostly interest?
Answers: 1) $90 2) $14,790 3) $2,000 4) The balance (and so the interest on it) is largest at the start.
Quick recap
- Amortization spreads a loan into equal payments of interest plus principal.
- An amortization schedule tracks each payment.
- Longer terms and higher rates increase total interest.