Loans and Amortisation
You borrow $400,000 to buy a house. Over 30 years, you will pay back nearly $800,000. Where does the extra $400,000 go? It goes to interest, and understanding exactly how it flows is the key to smarter financial decisions. Amortisation is the process of gradually paying off a loan through regular repayments, where each payment splits between interest and principal. Early payments are mostly interest; later payments are mostly principal.
Practise this lesson
Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
A $300,000 mortgage at 5% over 25 years costs about $1750/month. In the first year, roughly how much of your $21,000 in payments goes to interest vs principal? Predict before reading.
Before reading onwrite your gut feeling. We will revisit this at the end of the lesson.
Every amortisation problem uses the same four equations applied period by period.
Monthly repayment: $M = PV \times \dfrac{r}{1 - (1+r)^{-n}}$
Interest for period: $I = \text{Balance} \times r$
Principal for period: $P = M - I$
New balance: $\text{Balance}_{new} = \text{Balance}_{old} - P$
Key facts
- Amortisation formula
- Interest vs principal split
- Reducing balance concept
Concepts
- Why early payments are mostly interest
- How extra repayments save money
- The true cost of long-term loans
Skills
- Build amortisation schedules
- Calculate total interest
- Compare loan scenarios
An amortisation schedule shows how each repayment is split between interest and principal, and how the balance reduces over time.
Example: $20,000 car loan at 7.2% p.a. compounded monthly over 3 years.
$r = 0.072 \div 12 = 0.006$, $n = 36$. Using the repayment formula:
$$M = 20000 \times \frac{0.006}{1 - (1.006)^{-36}} = \$618.96 \text{ per month}$$| Month | Opening Balance | Interest | Principal | Closing Balance |
|---|---|---|---|---|
| 1 | $20,000.00 | $120.00 | $498.96 | $19,501.04 |
| 2 | $19,501.04 | $117.01 | $501.95 | $18,999.09 |
| 3 | $18,999.09 | $113.99 | $504.97 | $18,494.12 |
| … | … | … | … | … |
| 36 | $615.27 | $3.69 | $615.27 | $0.00 |
An amortisation schedule shows each repayment split into interest (= r × opening balance) and principal reduction (= repayment − interest). Opening balance for next period = previous balance − principal paid. Final balance = 0.
Pause, copy the four amortisation columns: opening balance, interest (= r × opening balance), principal paid (= repayment − interest), and closing balance (= opening − principal paid) into your book.
Quick check: In Month 1 of a $20,000 loan at 7.2% p.a. compounded monthly (r = 0.006), the interest component of the repayment is:
The amortisation schedule shows that each repayment splits into interest (= r × opening balance) and a principal reduction. When you make an extra payment, every dollar above the scheduled repayment goes directly to reducing the principal, which shrinks every future interest charge, shortening the loan and saving money that compounds over the remaining term.
Adding even small extra amounts to your regular repayment dramatically reduces total interest and loan term.
Example: $400,000 mortgage at 4.8% p.a. compounded monthly over 30 years.
$r = 0.004$, $n = 360$.
$$M = 400000 \times \frac{0.004}{1 - (1.004)^{-360}} = \$2098.64 \text{ per month}$$Total payments = $2098.64 × 360 = $755,510. Total interest = $355,510.
If you pay $2,200/month instead (only $101.36 extra per month):
- New term ≈ 25 years, saves 5 years of repayments
- Total interest ≈ $260,000, saves approximately $95,000
Extra repayments directly reduce the principal, which reduces future interest charges. Total interest saved = interest that would have accrued on the extra amount over remaining periods. Even small extra payments in early periods save significant interest.
Pause, copy the principal-reduction mechanism (every extra dollar goes directly to principal, cutting all future interest charges) and note why early extra payments save the most: the principal reduction compounds over more remaining periods into your book.
True or false: Paying $100 extra per month on a 30-year mortgage saves more money if you start in Year 1 than if you start in Year 20.
Worked examples · reveal each step
$250,000 mortgage at 5.4% p.a. compounded monthly over 20 years. Find the monthly repayment, total interest, and remaining balance after 5 years.
$15,000 personal loan at 9.6% p.a. compounded monthly over 5 years. Find monthly repayment, total interest, and savings from paying $50 extra per month.
Extra repayments save interest because they reduce the balance on which future interest is calculated. When comparing two loans, different rates, terms, or repayment structures, the definitive comparison is total interest paid: total repayments (repayment × n) minus the principal borrowed. A lower interest rate does not always mean lower total cost if the term is longer.
When comparing loans, always consider:
- Interest rate: Lower is better, but watch for fees.
- Fees: Establishment fees, monthly account fees, early repayment penalties.
- Features: Offset account, redraw facility, repayment flexibility.
- Comparison rate: Includes most fees, the fairest basis for comparison.
Example: Loan A: 5.0% with $0 fees. Loan B: 4.8% with $10/month fee on $300,000 over 25 years.
Lower rate on Loan B saves ≈ $150/year on interest, but pays $120/year in fees. Net saving = only $30/year. Loan A is very slightly better despite the higher headline rate.
To compare two loans: calculate total repayments for each (monthly repayment × number of repayments), then subtract the principal to find total interest. The loan with lower total interest is cheaper, regardless of the rate label.
Pause, copy total interest = (repayment × n) − principal, and the comparison rule: the cheaper loan is the one with lower total interest, not necessarily the one with the lower advertised rate into your book.
Fill the gap: For a $20,000 car loan at 7.2% p.a. compounded monthly over 3 years, the monthly repayment is $618.96. The total interest paid over the 36 months is $ .
Common errors · the 3 traps that cost marks
Match each term to its meaning:
Quick-fire practice · 2 activities
Create the first 4 months of an amortisation schedule for a $30,000 loan at 6% p.a. compounded monthly over 4 years. For each month, show opening balance, interest, principal, and closing balance.
Compare total interest on a $300,000 mortgage at 5% p.a. compounded monthly for 25 years vs 30 years. How much extra does the longer term cost?
Top 3 list: Name THREE strategies a borrower can use to reduce the total interest paid on a mortgage. For each, explain the mathematical reason why it works.
Monthly interest in Month 1 = $300,000 × 0.05/12 = $1,250. Monthly payment ≈ $1,750. So in Month 1, interest = $1,250 and principal = only $500. Over the first year, interest totals about $14,500 and principal about $6,500. This means roughly 69% of your first year's payments go to interest. This is why making extra repayments early is so powerful, every extra dollar reduces the balance that next month's $1,250 interest is calculated on.
What has changed in your understanding? What did you get right? What surprised you?
Pick your answer, then rate your confidencethat tells the system what to drill next.
Q1. A $20,000 car loan at 7.2% p.a. compounded monthly has r = 0.006 and n = 36. Which expression gives the correct monthly repayment?
Q2. In Month 1 of a $30,000 loan at 6% p.a. compounded monthly, the interest component of the repayment is:
Q3. The total interest paid on a loan is most correctly calculated as:
Q4. Over the life of a mortgage with fixed monthly repayments, the interest component of each repayment:
Q5. A comparison rate is most useful for borrowers because it:
SA 1. A $350,000 mortgage at 5.4% p.a. compounded monthly over 25 years. (a) Calculate the monthly repayment. (b) Create an amortisation schedule for the first 3 months. (c) Calculate the total interest over the life of the loan. (2 marks)
SA 2. A couple has a $450,000 mortgage at 4.8% p.a. compounded monthly over 30 years. They can afford to pay $300 extra per month. (a) Find the original monthly repayment. (b) Find the new loan term with the extra payment. (c) Calculate the total interest saved. (2 marks)
SA 3. (a) Explain mathematically why paying fortnightly (half the monthly amount, 26 times per year) reduces total interest compared to monthly payments. (b) For a $500,000 mortgage at 5% p.a. compounded monthly over 30 years, calculate the approximate saving from switching to fortnightly payments. (c) A bank offers a 0.2% rate discount but charges a $395 annual fee. For a $400,000 loan over 25 years, is this deal worthwhile? Show all working. (3 marks)
Comprehensive answers (click to reveal)
MC 1, B: The repayment formula is M = PV × r / [1−(1+r)^−n].
MC 2, A: Interest = 30000 × 0.005 = $150.00.
MC 3, C: Total interest = M×n − PV. Other options use simple interest on the original balance, which ignores the reducing balance.
MC 4, D: As the balance falls each period, less interest is charged, so more of the fixed repayment reduces principal.
MC 5, B: The comparison rate includes fees, it is the fairest basis for comparing loan products.
SA 1 (2 marks): (a) r=0.0045, n=300. M = 350000×0.0045/[1−(1.0045)^−300] = 1575/0.7387 = $2132.14/month [0.5 mark]. (b) Month 1: OB=350000, I=1575.00, P=557.14, CB=349442.86. Month 2: OB=349442.86, I=1572.49, P=559.65, CB=348883.21. Month 3: OB=348883.21, I=1569.97, P=562.17, CB=348321.04 [1 mark]. (c) Total = 2132.14×300 = $639,642. Interest = $289,642 [0.5 mark].
SA 2 (2 marks): (a) M = 450000×0.004/[1−(1.004)^−360] = 1800/0.7624 = $2360.97/month [0.5 mark]. (b) New M = $2660.97. Solve: [1−(1.004)^−n] = 450000×0.004/2660.97 = 0.6764. (1.004)^−n = 0.3236. n = ln(0.3236)/−ln(1.004) ≈ 282 months = 23.5 years [0.5 mark]. (c) Original total = 2360.97×360 = $849,949. Interest = $399,949. New total = 2660.97×282 = $750,394. Interest = $300,394. Saved ≈ $99,555 [1 mark].
SA 3 (3 marks): (a) 26 half-payments = 13 full payments per year instead of 12. The extra annual payment goes entirely to principal, reducing the balance faster. Paying more frequently also means slightly less interest accrues between payments [1 mark]. (b) Monthly: M = $2684.11, total = $966,279, interest = $466,279. Fortnightly ≈ 22.5 years, total interest ≈ $340,000. Saved ≈ $126,000 [1 mark]. (c) At 5%: M=$2338.36, total=$701,508. At 4.8%: M=$2295.25, total=$688,575. Repayment saving = $12,933. Fees = 25×$395 = $9,875. Net saving = $3,058. Yes, worthwhile [1 mark].
Five timed questions on amortisation, interest calculations, repayment schedules and loan comparisons. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
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