Model and analyse compound interest investments, reducing-balance loans, annuities and perpetuities using a first-order recurrence relation and a finance solver, and interpret balance, repayment, interest and the effect of changing parameters

What this dot point is asking

VCAA wants you to model real financial situations with a mixed first-order recurrence relation and then analyse them with a calculator finance solver. The four contexts are compound interest investments, reducing-balance loans, annuities (and the special case of perpetuities). You must set up the recurrence, find balances and repayments, compute total interest, and explain how changing the rate, repayment or term affects the outcome. This dot point is the climax of the Recursion and financial modelling core.

The mixed recurrence relation

A reducing-balance loan or an annuity follows

where $V_n$ is the balance after $n$ periods, $R$ is the per-period growth factor, and $D$ is the regular payment. Each period the balance grows by interest and then a payment is made.

• Compound interest investment with no payments: $D = 0$, so $V_{n+1} = R\,V_n$ (purely geometric growth).
• Reducing-balance loan: $V_0$ is the amount borrowed, $D$ is the repayment, and the balance falls toward zero.
• Annuity: $V_0$ is the amount invested, $D$ is the withdrawal, and the balance falls as you draw down.
• Perpetuity: the payment exactly equals the interest each period, so the balance never changes: $D = \dfrac{r}{100\,k}\,V_0$.

Compound interest

For an investment of principal $P$ at annual rate $r$ percent compounded $k$ times per year for $t$ years, the value is

The interest earned is $A - P$. Increasing the compounding frequency $k$ slightly increases the final value because interest is added more often.

Reducing-balance loans on the finance solver

Most loan questions are solved with the calculator finance solver, which links five quantities: $N$ (number of payments), $I\%$ (annual interest rate), $PV$ (present value or amount borrowed), $PMT$ (regular payment), and $FV$ (future value, usually $0$ when a loan is fully repaid). Enter the four you know with the correct sign convention (money received is positive, money paid is negative) and solve for the fifth.

Effect of changing parameters

You must be able to explain, in words, what happens when a parameter changes. Raising the interest rate increases the total interest and either lengthens the term or raises the required repayment. Increasing the repayment shortens the term and reduces total interest. Increasing the compounding frequency slightly increases the cost of a loan or the value of an investment.

Why this matters for the exams

Financial modelling is examined heavily and rewards method marks: show the recurrence, the per-period factor and the finance-solver inputs even when a calculator gives the number. Exam 1 tests single-step recurrence and compound-interest reasoning by hand; Exam 2 expects efficient solver use to find a payment, a term or a final balance and a clear interpretation in dollars.