Mixed Financial Problems

For mixed financial problems, follow these five steps:

1. Identify what you need to find: total cost, monthly payment, time to goal, or a comparison between options.
2. List all given information: principal, rates, time periods, compounding frequency, fees.
3. Choose the right formula(s): match the situation — is money growing (compound interest / FV annuity) or is money being borrowed and repaid (loan repayment / PV annuity)?
4. Calculate each component separately: work step by step, keeping full precision until the final answer.
5. Compare and conclude: state which option is cheaper or better, and give a brief justification.

Common combinations that appear in HSC questions:

• Compare saving vs borrowing (FV of annuity vs total repayments on a loan)
• Evaluate refinancing (compare total costs before and after, including fees)
• Salary sacrifice vs after-tax investing (tax-adjusted returns)
• Multiple debts — which to pay first (debt avalanche method)

Problem: You need a $25,000 car. You can save $600/month at 4.8% p.a. compounded monthly, or take a 5-year loan at 7.2% p.a. compounded monthly.

Saving option:

We need $FV = 25000$. Using $FV = M \times \frac{(1+r)^n - 1}{r}$ with $M = 600$ and $r = \frac{0.048}{12} = 0.004$:

$$25000 = 600 \times \frac{(1.004)^n - 1}{0.004}$$ $$\frac{25000 \times 0.004}{600} = (1.004)^n - 1 \implies (1.004)^n = 1.1\overline{6}$$ $$n = \frac{\ln 1.1\overline{6}}{\ln 1.004} \approx 38.7 \text{ months (round up to 39)}$$

Total contributed = $600 \times 39 = \$23{,}400$ (the rest is interest earned). You reach $25,000 in just over 3 years.

Borrowing option:

$r = \frac{0.072}{12} = 0.006$, $n = 60$.

$$M = 25000 \times \frac{0.006}{1 - (1.006)^{-60}} = \frac{150}{0.30265} \approx \$495.03 \text{ per month}$$

Total repayments = $495.03 \times 60 = \$29{,}702$. Interest paid = $\$4{,}702$.

Comparison: Saving costs $0 interest but takes 3.25 years — you cannot drive the car now. Borrowing gives immediate access but costs $4,702 in interest. The right choice depends on whether you need the car immediately and whether you can tolerate the wait.

Problem: A 30-year-old can salary sacrifice $500/month into super (taxed at 15%) earning 7% p.a. compounded monthly, OR invest $500/month after-tax in shares earning 8% p.a. compounded monthly. Marginal tax rate = 32.5%.

Super option: Contribution is pre-tax. $M = 500$, $r = 7/12\% = 0.005833$, $n = 420$ months (35 years).

$$FV = 500 \times \frac{(1.005833)^{420} - 1}{0.005833} \approx 500 \times 3748 = \$1{,}874{,}000$$

Shares option: After $500 is taxed at 32.5%, only $500 \times (1 - 0.325) = \$337.50$ is invested. Returns are also taxed, so effective return $\approx 8\% \times (1 - 0.325) = 5.4\%$. $r = 0.0045$, $n = 420$.

$$FV = 337.50 \times \frac{(1.0045)^{420} - 1}{0.0045} \approx 337.50 \times 1924 = \$649{,}000$$

Difference: Super wins by approximately $\$1.2$ million, entirely due to tax concessions on contributions and earnings.