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Loans & Amortization

lesson · about 3 minutes

Monthly interest, how each payment splits, and what a loan really costs.

Pen and paper is fine · no calculator needed why?

Opens at level 34. You're level 1. You can read and practice here now.

the idea

A loan payment does two jobs. First it pays this month's interest: the balance times the monthly rate. The rest of the payment reduces the principal, the amount you still owe. Paying a loan down this way, month by month, is called amortization.

Interest is charged on the balance, so it shrinks as the balance shrinks, and later payments pay off more principal.

Over the whole loan, the interest you pay is everything you repay minus what you borrowed.

techniques

Interest first, then principal

One monthly payment on a loan.

Interest is the balance × the monthly rate. 1.5% is 0.015.
Principal repaid is the payment minus the interest.
The new balance is the old balance minus the principal repaid.

worked example

Example: A bakery's loan begins the month at $12,000. Assume interest is 1% for the month, and a payment of $500 is made at month-end. What is the ending principal balance?

Interest: $12,000 × 0.01 = $120.
Principal repaid: $500 − $120 = $380.
Ending principal: $12,000 − $380 = $11,620.

Answer: $11,620

Repeat the month

Two payments in a row.

Add this month's interest to the balance, then take off the payment.
Start the next month from that new balance.
Work out the interest again. It is smaller now.

worked example

Example: You owe $5,000 on a loan. Assume it charges 2% interest per month: each month, interest is added on the balance, then your $500 payment comes off. What is the balance after two payments?

Month 1: $5,000 + $100 − $500 = $4,600.
Month 2: $4,600 + $92 − $500 = $4,192.

Answer: $4,192

Total repaid minus borrowed

Total interest over a loan with equal payments.

Multiply the payment by the number of payments.
Take away the amount borrowed. What remains is the interest.

worked example

Example: You borrow $6,000 and repay it with 24 monthly payments of $275. How much interest do you pay in total?

24 × $275 = $6,600.
$6,600 − $6,000 = $600.

Answer: $600

watch out for

Taking the whole payment off the principal. Part of it pays interest, so the balance falls by less than the payment.
Charging month 2's interest on the original balance instead of the smaller balance after month 1.
Giving the total repaid as the interest.
Moving the decimal one place too few: 0.75% is 0.0075, not 0.075.

practice

This month's interest

worked example

A loan's balance is $53,000. Assume it charges 2% interest per month. What is this month's interest?

Answer: $1,060.00

2% = 0.02, so $53,000 × 0.02 = $1,060.

Balance after one payment

worked example

A loan begins the month at $44,000. Assume interest is 0.5% for the month, and a payment of $250 is made at month-end. What is the ending principal balance?

Answer: $43,970.00

Interest = $44,000 × 0.005 = $220.
Principal repaid = $250 − $220 = $30.
Ending principal = $44,000 − $30 = $43,970.

Balance after two payments

worked example

You owe $18,000 on a loan. Assume it charges 1% interest per month: each month, interest is added on the balance, then your $2,200 payment comes off. What is the balance after two payments?

Answer: $13,939.80

Month 1: $18,000 + $180 − $2,200 = $15,980.
Month 2: $15,980 + $159.80 − $2,200 = $13,939.80.
Interest shrinks as the balance shrinks.

Total interest over the loan

worked example

You borrow $5,000 and repay it with 12 monthly payments of $426. How much interest do you pay in total?

Answer: $112.00

Total paid = 12 × $426 = $5,112.
Interest = $5,112 − $5,000 = $112.