Loan Repayments · Learning My Way

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🧰 Foundations: Percentages (R = 1+r)Recurrence Relations Compound Interest

Make sure you've done Reducing Balance Loans from Unit 3 first. This page assumes you know the basic recurrence relation for a loan. Here we extend it to construct full amortisation schedules and calculate total interest.

How each repayment is structured

Each loan repayment does two things: it covers the interest charged that period, and whatever is left over reduces the outstanding loan principal. Early repayments are mostly interest, later ones are mostly principal.

Each repayment period Interest charged = Opening balance × monthly rate
Closing balance = Opening balance + Interest − Repayment
Principal reduced = Repayment − Interest charged
Monthly rate = annual rate ÷ 12

Monthly rate example: A loan at 12% p.a. → monthly rate = 12% ÷ 12 = 1% = 0.01 per month.
A loan at 6% p.a. → monthly rate = 6% ÷ 12 = 0.5% = 0.005 per month.

Building an amortisation table

Worked Example A · constructing the schedule

A $20,000 loan at 12% p.a. (1% per month). Monthly repayment is $500. Construct the first 4 months of the amortisation table.

Month	Opening balance	Interest (×1%)	Repayment	Closing balance
1	$20,000	$200	$500	$19,700
2	$19,700	$197	$500	$19,397
3	$19,397	$194	$500	$19,091
4	$19,091	$191	$500	$18,782

Notice: the interest charged decreases each month because the balance is falling. This means more of each repayment goes toward reducing principal over time.

Calculating total interest paid

The total interest paid over the life of a loan is simply the difference between everything you pay out and the original loan amount.

Total Interest Total interest = Total repayments − Original loan amount
Total repayments = Repayment amount × Number of repayments

Worked Example B · total interest

A $12,000 loan is fully repaid with 12 monthly payments of $1,100. How much total interest was paid?

Total repayments:12 × $1,100 = $13,200

Total interest:$13,200 − $12,000 = $1,200

The effect of extra repayments

Making extra repayments above the minimum reduces the outstanding balance faster. A smaller balance means less interest is charged next month, so the savings compound over time.

Worked Example C · extra repayment effect

A loan has an opening balance of $15,000 at 12% p.a. (1%/month). The minimum repayment is $500. This month the borrower pays an extra $200. What is the closing balance?

Interest:$15,000 × 0.01 = $150

Total paid:$500 (min) + $200 (extra) = $700

Closing balance:$15,000 + $150 − $700 = $14,450

Without the extra $200, closing balance would have been $14,650. That $200 extra saves interest in every subsequent month.

Without extra repayment

Opening: $15,000
Interest: $150
Repayment: $500
Closing: $14,650

With $200 extra

Opening: $15,000
Interest: $150
Repayment: $700
Closing: $14,450

Practice Questions

1. A loan has an opening balance of $20,000 at 12% p.a. (1% per month). What is the interest charged in the first month? ▶

Interest = $20,000 × 0.01 = $200

2. Using Q1 (opening $20,000, interest $200), the monthly repayment is $500. What is the closing balance at the end of month 1? ▶

Closing = Opening + Interest − Repayment
= $20,000 + $200 − $500 = $19,700

3. A borrower makes a monthly repayment of $500 for 24 months. What are the total repayments? ▶

Total repayments = 24 × $500 = $12,000

4. A $12,000 loan is repaid with 12 monthly payments of $1,100. How much total interest was paid? ▶

Total repayments = 12 × $1,100 = $13,200
Total interest = $13,200 − $12,000 = $1,200

5. A loan's opening balance is $8,500 at 12% p.a. (1% per month). The monthly repayment is $400, plus an extra $100 this month. Find the closing balance. ▶

Interest = $8,500 × 0.01 = $85
Total repaid = $400 + $100 = $500
Closing = $8,500 + $85 − $500 = $8,085

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🏦 Loan Repayments