Depreciation and Loans
Straight-line and declining-balance depreciation, comparing the two methods, reducing-balance loans modelled in a table, the effect of extra or more frequent repayments, credit card interest compounded daily, fees, minimum payments, and choosing between a credit card and a personal loan.
Contents
1.What you need to know
This focus area is about money going down: things losing value, and the cost of borrowing. Most questions are practical: a car, a work vehicle, a credit card statement, a small loan.
- Straight-line depreciation: .
- Declining-balance depreciation: .
- Compare the two methods numerically or from a graph, and model them in a spreadsheet.
- Reducing-balance loans: model a loan in a table for up to three periods, and explain the effect of changing the repayment, the repayment frequency, or making a lump-sum payment.
- Credit cards: interest-free periods, fees, minimum payments, interest compounded daily, reading a statement, and comparing a credit card with a personal loan.
2.Depreciation
Depreciation is the loss in value of an item over time, such as a car, a tractor or a laptop. The value at a later time is the salvage value .
Straight-line depreciation
The item loses the same dollar amount each period.
- = initial value, = depreciation amount per period, = number of periods.
The graph of value against time is a straight line sloping down, with gradient . The value can reach $0.
Declining-balance depreciation
The item loses the same percentage of its current value each period.
- = depreciation rate per period, as a decimal.
The graph is a curve that drops steeply at first and then flattens out. The value never reaches $0. This formula is the compound interest formula with instead of : compound interest grows, declining balance shrinks.
Worked example: comparing the two methods
A ute costs $22 000. Method 1 is straight-line with per year. Method 2 is declining-balance at 25% p.a.
| Year | Straight-line | Declining-balance |
|---|---|---|
| 0 | $22 000 | $22 000 |
| 1 | $18 500 | $16 500 |
| 2 | $15 000 | $12 375 |
| 3 | $11 500 | $9281.25 |
| 4 | $8000 | $6960.94 |
| 5 | $4500 | $5220.70 |
Declining balance loses more in the early years ($5500 in year 1, compared with $3500), so its value is lower at first. Its yearly loss shrinks, though, and by year 5 the straight-line value has dropped below it. The graphs cross between years 4 and 5.
Declining balance often matches real cars better: a new car loses a lot of value in its first year.
Spreadsheets
In a spreadsheet, straight-line depreciation subtracts the same amount each row (for example =B2-3500), and declining balance multiplies by the same factor (=C2*0.75). Fill down to see the value each year.
3.Loans
Reducing-balance loans
Most loans (home loans, car loans, personal loans) are reducing-balance loans. Each period:
- Interest is calculated on the balance owing: interest = balance × rate per period.
- The interest is added to the balance.
- The repayment is subtracted.
Because the balance falls, each period's interest is smaller, so more of each repayment goes towards the principal.
Worked example: a loan in a table
Sam borrows $4000 at 12% p.a., with interest calculated monthly (1% per month). He repays $1200 at the end of each month. Interest is rounded to the nearest cent.
| Month | Opening balance | Interest | Balance + interest | Closing balance |
|---|---|---|---|---|
| 1 | $4000.00 | $40.00 | $4040.00 | $2840.00 |
| 2 | $2840.00 | $28.40 | $2868.40 | $1668.40 |
| 3 | $1668.40 | $16.68 | $1685.08 | $485.08 |
After 3 months Sam owes $485.08. The interest so far is dollars. Notice the interest falls each month because the balance is smaller.
Changing the repayments
| Change | Effect on the loan |
|---|---|
| Bigger repayments | balance falls faster: shorter term, less total interest |
| Extra repayments | same as above; even small extras save a lot over a long loan |
| A lump-sum payment | the balance drops at once, so all later interest is smaller |
| More frequent repayments (fortnightly instead of monthly) | the balance is reduced sooner, so slightly less interest; paying half the monthly amount each fortnight also gives 26 half-payments a year, one extra month's worth |
Total cost of a loan = the total of all repayments (plus any fees). Interest paid = total cost − amount borrowed.
Spreadsheets and loan calculators
For loans over many years, a spreadsheet or online calculator is used: each row calculates the interest, the new balance and the repayment. In the exam you may be given part of such a spreadsheet and asked to fill in a row or interpret the results.
4.Credit cards
A credit card is a reducing-balance loan where you choose how much to borrow and when to repay, as long as you make the minimum payment each month.
Key terms
- Interest-free period: a set number of days (often up to about 55) when no interest is charged on purchases, if the full closing balance is paid by the due date.
- Closing balance: the amount owing at the end of the statement period.
- Minimum payment: the smallest amount you must pay by the due date, often a small percentage of the closing balance (such as 2%) or a set amount, whichever is greater.
- Fees: an annual fee, late payment fees, cash advance fees (cash withdrawals usually have no interest-free period and a higher rate), and foreign transaction fees.
Credit card interest rates are much higher than those for most other loans: often around 20% p.a., which is several times a typical home loan rate and above most personal loan rates.
Interest compounded daily
Interest on a credit card is usually calculated daily and compounds:
where is the yearly rate and is the number of days.
Example. $1850 is owed on a card charging 18.99% p.a., compounded daily, for 31 days.
The interest is about $30.07. Simple interest would give a slightly smaller answer, because it ignores interest on interest.
Only paying the minimum
If you pay only the minimum, most of the balance stays and keeps attracting interest. With a 2% minimum and an interest rate near 20% p.a., the balance falls very slowly and it can take many years, with interest costing more than the original purchases. Statements in Australia must show how long it would take to repay the balance with minimum payments only.
Credit card or personal loan?
| Credit card | Personal loan | |
|---|---|---|
| Interest rate | high | lower |
| Flexibility | borrow and repay any time | fixed amount, fixed repayments |
| Best for | short-term purchases paid off within the interest-free period | medium-term goals (a car, a repair) repaid over 1 to 5 years |
To compare, work out the total amount repaid for each option, including fees.
5.Common mistakes
- Using instead of for declining balance.
- Giving the depreciation amount when the question asks for the salvage value (or the reverse).
- Using the yearly interest rate for each month in a loan table: divide by 12.
- In a loan table, subtracting the repayment before adding the interest when the question says interest is added first.
- Forgetting fees when comparing the cost of loans.
- Choosing the smaller option in a "whichever is greater" minimum payment rule.
6.Exam tips
- Write the formula first and substitute clearly, even when it is given in the question or on the reference sheet.
- In a loan table, keep the columns in order and round interest to the nearest cent each row if the question says to.
- For compare questions, give numbers for both options and a concluding sentence.
- For explain questions about credit cards, use the key terms: interest-free period, minimum payment, compounded daily, fees.
7.Links to other topics
- Investment (Year 12): compound interest has the same structure as declining-balance depreciation and credit card interest.
- Algebraic relationships (Year 12): straight-line depreciation is a linear model and declining balance is an exponential decay model.
- Managing money (Year 11): buying on terms, budgets and percentage decrease.
Test yourself
Practise 12 exam-style questions on Depreciation and Loans, each with instant marking and a worked solution.
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