In practice, loan interest rates change due to central bank policy, and borrowers may choose to change repayment amounts (e.g., lump sum payments, increased regular repayments). VCAA General Mathematics requires analysis of these scenarios using recurrence relations.
A reducing balance loan with principal $P$, periodic interest rate $r$ (per period), and repayment $d$ per period follows:
$$A_{n+1} = A_n(1 + r) - d, \quad A_0 = P$$
The loan is repaid when $A_n = 0$ (or $A_n \leq 0$).
A borrower makes an extra lump sum payment of $L$ after $k$ periods.
New balance after lump sum: $A_k^{\text{new}} = A_k - L$
Continue the recurrence from $A_k^{\text{new}}$ with the same $r$ and $d$ to find the new repayment period.
Original loan: \$20,000, interest 6% p.a. compounded monthly (0.5% per month), monthly repayment \$400.
After 12 months (using CAS), balance $A_{12} \approx \$18{,}428$.
A lump sum of \$3,000 is paid: new balance $= \$15{,}428$.
The borrower continues \$400/month. The loan is now paid off several months earlier.
After $k$ periods at rate $r_1$, the rate changes to $r_2$.
New recurrence from period $k$: $A_{n+1} = A_n(1 + r_2) - d$
If the borrower wants to maintain the same repayment period, the repayment $d$ must be recalculated.
For a loan balance $B$, rate $r$ per period, and $m$ remaining periods:
$$d = \frac{Br(1+r)^m}{(1+r)^m - 1}$$
(This is found using CAS Finance functions or by solving the recurrence.)
Increasing $d$ reduces the principal faster, saves total interest, and shortens the loan term.
Decreasing $d$ has the opposite effect — and if $d < Ar$ (payment less than interest charged), the loan never reduces.
Minimum repayment condition: $d > Ar$ (repayment must exceed interest charged in that period).
| Change | Effect |
|---|---|
| Increase repayment | Shorter loan term, less total interest |
| Lump sum payment | Immediate reduction in principal |
| Rate increase | Longer term or higher required repayment |
| Rate decrease | Shorter term or lower repayment possible |
VCAA FOCUS: Exam questions often describe a rate change mid-loan and ask you to find the new required repayment or revised pay-off time. Set up the recurrence carefully with the new values.
COMMON MISTAKE: Forgetting to update the interest rate in the recurrence relation when the rate changes. The recurrence must use the new $r$ from the point of change onward.
