In a reducing balance loan (also called a declining balance loan), interest is calculated on the outstanding (current) balance each period. As repayments are made, the balance reduces, and so does the interest charged.
This is how most home loans, car loans, and personal loans work.
$$V_{n+1} = R \cdot V_n - d, \quad V_0 = L$$
Where:
- $V_n$ = outstanding balance after $n$ periods
- $R = 1 + \frac{r}{100}$ (compound factor)
- $d$ = repayment per period
- $L$ = initial loan amount (principal)
- $r$ = interest rate per period (%)
Loan: \$20,000 at 6% p.a. compounded monthly, monthly repayment \$500.
| Month ($n$) | Balance ($V_n$) | Interest charged | Repayment | Principal reduced |
|---|---|---|---|---|
| 0 | 20,000.00 | — | — | — |
| 1 | 19,600.00 | 100.00 | 500.00 | 400.00 |
| 2 | 19,198.00 | 98.00 | 500.00 | 402.00 |
| 3 | 18,794.09 | 96.09 | 500.00 | 403.91 |
Observation: Each month, more of the repayment goes toward principal and less toward interest.
$$\text{Interest in period } n = V_{n-1} \times \frac{r}{100}$$
Use CAS / TVM solver:
- Enter: $PV$, $I\%$, $PMT$, $N$
- Solve for $FV$ (remaining balance)
For the balance to decrease, the repayment must exceed the interest charged:
$$d > V_0 \times \frac{r}{100}$$
Example: For a \$20,000 loan at 0.5% monthly, minimum to reduce balance: \$0.005 \times 20000 = \$100$. Any repayment above \$100 reduces the balance.
Use TVM solver with $FV = 0$:
- $N, PV, I\%$ known
- Solve for $PMT$
$$\text{Total interest} = (d \times n) - L$$
where $n$ = total number of repayments and $L$ = original loan.
KEY TAKEAWAY: A reducing balance loan uses $V_{n+1} = RV_n - d$. Interest is charged on the current balance, so early repayments mainly cover interest while later repayments mainly reduce principal.
EXAM TIP: VCAA commonly asks for the balance after a specific number of payments, or the repayment needed to pay off the loan by a set time. Use TVM solver and show the inputs clearly.
COMMON MISTAKE: Using the annual interest rate directly instead of the per-period rate. Monthly loans need a monthly rate: divide annual rate by 12.
