Real financial situations rarely have a constant interest rate over the entire life of a loan or investment. Interest rates may rise, fall, or be fixed for a period then change. Repayment amounts may also be altered. This section explores the impact of such changes.
Strategy: Split the loan into segments. The balance at the end of one segment becomes $V_0$ for the next segment with the new rate.
Loan: \$50,000 at 5% p.a. monthly for 3 years, then rate rises to 7% p.a. Monthly repayment stays at \$1000.
Segment 1 (first 36 months at 5% p.a.):
- Monthly rate: 0.4167%
- $V_{n+1} = 1.004167 \times V_n - 1000$, $V_0 = 50000$
- Find $V_{36}$ using CAS → $V_{36} = \$42,816$ (approx)
Segment 2 (from month 37, rate = 7% p.a.):
- Monthly rate: 0.5833%
- New recurrence: $V_{n+1} = 1.005833 \times V_n - 1000$, $V_0 = 42816$
The loan now takes longer to pay off and more total interest is paid.
Increasing the repayment amount:
- Reduces the outstanding balance faster
- Reduces the total interest paid
- Shortens the loan term
| Monthly repayment | Time to pay off \$20,000 at 6% p.a. monthly |
|---|---|
| \$400 | ~64 months |
| \$500 | ~48 months |
| \$600 | ~38 months |
Observation: Each \$100 extra per month saves significantly more than 10 months — interest savings compound.
Decreasing the repayment (or missing a payment):
- Balance decreases more slowly
- More total interest paid
- If repayment < interest charged, balance increases (negative amortisation)
A lump sum extra payment directly reduces the principal, lowering all future interest charges:
- Update $V_n$ at the time of the lump sum: $V_n^{\text{new}} = V_n - \text{lump sum}$
- Continue with the same recurrence relation from the new $V_n$
For savings annuities:
- Higher interest rate → balance grows faster
- Higher regular deposit → larger future value
- Longer investment term → significantly larger final balance (due to compounding)
| Monthly deposit | Interest rate | FV after 10 years |
|---|---|---|
| \$200 | 4% p.a. | \$29,452 |
| \$200 | 6% p.a. | \$32,776 |
| \$300 | 6% p.a. | \$49,164 |
| Change | Loan | Investment |
|---|---|---|
| ↑ Interest rate | Slower payoff, more interest | Faster growth |
| ↑ Repayment/deposit | Faster payoff, less interest | Larger FV |
| Lump sum payment | Reduces balance immediately | — |
| ↑ Term | — | Much larger FV (compounding) |
KEY TAKEAWAY: Small changes in interest rates or repayment amounts have large cumulative effects over time due to compounding. Using CAS to model “what if” scenarios is essential.
EXAM TIP: VCAA may give a scenario where the interest rate changes part way through a loan. Always recalculate the new balance at the change point, then restart the recurrence relation with the new rate and the updated balance as $V_0$.
VCAA FOCUS: Being able to explain the direction of the effect (increases balance / reduces term / saves interest) and quantify it using technology are both assessed.
