Borrow $10,000 for a truck and you pay back more than $11,000. Where does the extra $1,297.52 go, and why is the very first month the most expensive one?
Learning objectives. After this week you can (1) turn an annual rate into a monthly one; (2) find the monthly payment on a loan; and (3) split any payment into interest and principal.
Remember this: every payment pays this month's interest first. Whatever is left shrinks the loan.
1) Financing the $10,000 truck
Monthly rate i = Annual rate ÷ 12
A new truck costs $10,000. Instead of paying cash, you could borrow the principal (the amount borrowed) and pay it back in equal monthly payments. Rates are quoted per year but charged per month, so a 12% APR is 12 ÷ 12 = 1% per month, i = 0.01.
2) The payment formula
Payment = Principal × i ÷ (1 − (1 + i)−n) where n = number of payments
$10,000 at 1% for 24 months: 10,000 × 0.01 ÷ (1 − 1.01−24) ≈ $470.73 a month. You pay 24 × $470.73 = $11,297.52 in all, so the total interest is 11,297.52 − 10,000 = $1,297.52.
3) The amortization table
Interest = Balance × i · Principal paid = Payment − Interest · New balance = Balance − Principal paid
| Month | Payment | Interest (1%) | Principal | Balance |
|---|---|---|---|---|
| 0 | $10,000.00 | |||
| 1 | $470.73 | $100.00 | $370.73 | $9,629.27 |
| 2 | $470.73 | $96.29 | $374.44 | $9,254.83 |
| 3 | $470.73 | $92.55 | $378.18 | $8,876.65 |
The payment never changes, but the split does. Month 1 has the biggest interest charge because the balance is biggest. Each month a little more goes to principal, until the last payment brings the balance to $0.
Try it
A month that opens with a $7,500 balance, at 1% a month.
Interest = Balance × Rate
Fill in the boxes.
Worked example: should the loan scare you?
Turn the payment into a daily cost: $470.73 ÷ 30 ≈ $15.70 a day. Now put it next to what a truck earns. A truck making $250 a day in profit covers that payment many times over. A truck making $20 a day barely covers it. The loan is the same in both cases; the truck behind it is not.
Try it
Your payment is $470.73 and $100 of this month's went to interest.
Principal paid = Payment − Interest
Fill in the boxes.
4) Shorter loans and paying early
- Shorter term: the same loan over 12 months costs $888.49 a month, but total interest drops to $661.88. Bigger payments, less interest.
- Paying early: extra principal saves the interest you have not been charged yet. Add $100 to month 1 and the balance drops to $9,529.27. Interest already paid never comes back.
Quick self-check
A $6,000 loan at 1% a month has a payment of $282.44 over 24 months. (a) Month-1 interest? (b) Month-1 principal? (c) Balance after month 1? (d) Total interest? (Answers: $60.00; $222.44; $5,777.56; 24 × 282.44 − 6,000 = $778.56.)
Common mistakes
- Using 12% as the monthly rate. A 12% APR is 1% a month: i = 0.01, not 0.12.
- Charging interest on the original $10,000 every month. Interest is on what you still owe.
- Judging a loan by its payment alone. A lower payment over a longer term usually means more total interest.
This week in the arena
Build the first row of the table yourself: $10,000 at 1% with a $470.73 payment. Then open your season history and find your truck's typical daily net. Put the $15.70 a day next to it. How many times over would your truck cover the payment?