Model a perpetuity as a special annuity in which the regular payment equals the interest earned each period so the balance never changes, and find the perpetual payment, the required principal or the interest rate

What this dot point is asking

VCAA wants you to model a perpetuity: a lump sum invested so that the regular payment drawn out exactly equals the interest earned each period, leaving the balance unchanged forever. You recognise the constant-balance condition, use the relationship between payment, principal and interest rate, and solve for whichever quantity is unknown. Perpetuities fund scholarships and prizes, where a fixed amount must be paid out indefinitely without ever touching the principal.

The constant-balance condition

In a perpetuity the growth factor adds interest and the payment removes exactly that interest. The recurrence is the annuity form

but with $d$ chosen so that the interest added, $\frac{r}{100}V_n$, equals the payment $d$. Then the balance returns to its starting value every period:

The payment relationship

Because the payment equals the interest on the principal:

Rearranged, the principal needed to fund a given payment is $P = \dfrac{100\,d}{r}$, and the rate required is $r = \dfrac{100\,d}{P}$.

Matching the payment frequency to the rate

If the payment is made more often than yearly, use the rate per period. A perpetuity paying monthly at a nominal $6\%$ per annum uses $0.5\%$ per month, so $d = 0.005 \times P$. Always match the rate's period to the payment's period.

Why this matters for the exams

Perpetuity questions are short but reward the key insight that payment equals interest, so the balance is constant. They test the simple relationship $d = \frac{r}{100}P$ and its rearrangements, and link to the broader annuity model where the payment instead draws the balance down. State clearly that the balance is preserved, since that is the defining feature markers look for.