Model an annuity investment or savings plan where a regular payment is added to a compounding balance using the recurrence relation, analyse it on a finance solver, and find the final balance, total interest and required payment

What this dot point is asking

VCAA wants you to model a savings plan, called an annuity investment, where you add a regular payment to a balance that also earns compound interest. Each period the balance is multiplied by the growth factor and then a deposit is added. You set this up as a recurrence, analyse it on the finance solver, and find the final balance, the total interest earned, or the regular payment needed to reach a target. This is the savings mirror image of a reducing-balance loan.

The growth recurrence

Let $V_0$ be the initial deposit, $R = 1 + \frac{r}{100k}$ the growth factor per period (with $r$ the nominal annual rate and $k$ compounds per year), and $d$ the regular payment added each period. Then

This is a mixed recurrence: a geometric multiplication plus a constant addition. It is the same structure as a reducing-balance loan, except here the constant term is added rather than subtracted, because deposits build the balance up.

Using the finance solver

On a CAS finance solver you enter the number of payments $N$, the annual rate, the present value, the payment, the future value, the payments per year and the compounds per year. The sign convention matters: money you put into the investment (the present value deposit and the regular payments) is usually entered as negative, and the future value you receive is positive. Set the unknown field to solve.

Reading an amortisation-style table

VCAA frequently presents a savings plan as a partly completed table showing, for each period, the opening balance, the interest added, the payment in and the closing balance. To fill a missing cell you use the period structure: interest for a period equals the opening balance times the per-period rate $\frac{r}{100k}$; the closing balance equals opening balance plus interest plus deposit; and the next opening balance equals this closing balance. Working a single row by hand confirms you understand the model, and it is a common Exam 1 task where the finance solver is not allowed. Round only at the final step unless the question states otherwise, because rounding each interest figure early accumulates error across many periods.

Contrasting investments with loans

The annuity investment and the reducing-balance loan share the identical recurrence $V_{n+1} = R\,V_n \pm d$; the only difference is the sign of the constant term. In a savings plan the deposit is added ($+d$) so the balance climbs; in a loan the repayment is subtracted ($-d$) so the balance falls towards zero. This symmetry is worth stating explicitly because exam questions often pair the two, asking you to first build a deposit fund and then draw it down as a loan or pension. The interest interpretation also flips: for a savings plan interest is money you gain (balance minus principal minus deposits), whereas for a loan the total interest is what you pay (total repayments minus the amount borrowed).

Setting up the finance solver precisely

The seven solver fields must agree on the time unit. Set $N$ to the total number of payments, $I\%$ to the nominal annual rate, $PV$ to the negative of the initial deposit (money leaving you), $PMT$ to the negative of each regular deposit, $FV$ to the positive target balance, and both $P/Y$ and $C/Y$ to the number of periods per year. To solve for the required payment, leave $PMT$ blank and enter the target $FV$; to solve for the rate, leave $I\%$ blank. A sign error in $PV$ or $PMT$ is the single most common cause of a nonsensical solver answer, so check that money paid in is negative and the future value you receive is positive.

Solving for the required payment

A common question gives a savings target and asks for the regular deposit needed. You enter the target as the future value, the known present value and rate, and solve for the payment. The solver returns the deposit per period that reaches the goal.

Why this matters for the exams

Annuity investment questions appear most years and reward correct solver setup and a clean interest calculation. They contrast neatly with reducing-balance loans: same mixed recurrence, opposite direction of the constant term. Show the recurrence or the solver inputs explicitly, and state the total interest as balance minus principal minus deposits so the marker can follow your reasoning.