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.

Perpetuity as a special annuity

A perpetuity is the boundary case of a reducing-balance annuity. A general annuity follows $V_{n+1} = R\,V_n - d$, where the multiplier $R = 1 + \frac{r}{100}$ adds interest and the payment $d$ draws the balance down. In a perpetuity the payment is set exactly equal to the interest earned, $d = \frac{r}{100}V_n$, so the addition and the withdrawal cancel and the balance is fixed at the principal for every period. If the payment were even slightly smaller the balance would grow without limit; if it were slightly larger the balance would shrink and eventually run out, turning the perpetuity into an ordinary depleting annuity. Recognising this knife-edge condition is the key insight VCAA tests.

Cumulative payments and total payout

Because a perpetuity never stops, questions often ask not for a final balance but for the cumulative total of payments after $n$ periods. Since each payment is the constant $d = \frac{r}{100}P$, the running total after $n$ payments is simply $n\,d$, an arithmetic sum. To find when the cumulative payout first passes some target $T$, solve $n\,d > T$ for the smallest whole number $n$. This is exactly the structure of the 2025 multiple-choice item above: the payments form a linear (arithmetic) sequence even though the underlying account is compounding, because the balance never changes.

Matching technology to the model

On a CAS finance solver a perpetuity is entered with the present value and the future value equal in magnitude and opposite in sign (for example $PV = -200\,000$ and $FV = 200\,000$), the payment equal to the interest, and $N$ set to any convenient number of periods, since the balance is the same after every one. In Exam 1 (technology-free) you will not use the solver, so rely on the direct relationship $d = \frac{r}{100}P$ and its rearrangements. Always state, in words, that the balance is preserved because the payment equals the interest, as this is the reasoning mark examiners look for.

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.