Perpetuities: VCE General Mathematics Unit 3 Recursion and financial modelling

What this dot point is asking

A perpetuity is an investment that pays the same amount at the end of every period forever. The study design describes it as a "simple perpetuity as a special case of an annuity that lasts indefinitely". VCAA wants you to recognise the key condition (the payment exactly equals the interest earned, so the balance never changes), model it with the recurrence relation $V_{n+1} = RV_n - d$, and find whichever quantity is unknown: the payment, the principal, the interest rate, the value of $R$ or $d$ in the recurrence, or the total paid out over time.

Perpetuity questions are short, but the 2023 multiple-choice question on the value of $R$ was answered correctly by only $27\%$ of students, and the 2025 Examination 2 question on a constant loan balance by only $37\%$. The idea is simple; the algebra and the wording trip people up.

The answer

Why the balance never changes

An annuity starts with a lump sum $V_0$ that earns interest, and a payment $d$ is taken out each period:

where $r\%$ is the interest rate per period. The interest added in the first period is $\frac{r}{100}V_0$. There are three cases:

• $d$ greater than the interest: the balance falls and eventually runs out (an ordinary annuity);
• $d$ less than the interest: the balance grows;
• $d$ equal to the interest: the balance returns to exactly the same value every period, forever. This is a perpetuity.

For $200 000 at $5\%$ per annum paid annually, the interest is $0.05 \times 200\,000 = 10\,000$. With $d = 10\,000$:

and every later balance is also $200 000.

Finding the payment, principal or rate

The single relationship $d = \frac{r}{100}P$ answers most questions:

• Payment. $\$200\,000$ at $5\%$ per annum: $d = 0.05 \times 200\,000 = \$10\,000$ per year.
• Principal. A $3000 annual prize at $5\%$: $P = \frac{100 \times 3000}{5} = \$60\,000$.
• Rate. $2500 per month from $750 000: $r = \frac{100 \times 2500}{750\,000} = 0.3\overline{3}\%$ per month, which is $4\%$ per annum compounding monthly.

Matching the rate to the payment period

The rate must be the rate for one payment period. A perpetuity paying monthly at $4.5\%$ per annum compounding monthly uses $4.5 \div 12 = 0.375\%$ per month: $480 000 pays $0.00375 \times 480\,000 = \$1800$ per month. Using the annual rate by mistake gives a payment twelve times too big ($21 600 per month).

The recurrence relation

For a perpetuity written as $V_0 = a$, $V_{n+1} = RV_n - d$:

• the constant balance means $a = Ra - d$, so $R = \frac{a + d}{a}$ and $d = (R - 1)a$;
• $R - 1$ is the interest rate per period as a decimal: $V_{n+1} = 1.0125V_n - 4500$ with $V_0 = 360\,000$ has $1.25\%$ per quarter ($5\%$ per annum) and $d = 0.0125 \times 360\,000 = 4500$.

A recurrence with $d = 0$ is ordinary compound growth; a recurrence with $d$ equal to the interest is a perpetuity; a larger $d$ is an annuity that runs down.

Cumulative payments

Because every payment is identical, the total paid out after $n$ periods is simply $nd$, an arithmetic sequence, even though the account itself earns compound interest. Donald's $250 000 perpetuity at $5\%$ pays $12 500 a year; the total first exceeds $250 000 after 21 years (after 20 years it equals $250 000 exactly).

Using the finance solver

A perpetuity can be entered on the finance solver with $PV$ negative (money invested), $FV$ of the same size but positive (the principal is still there), and $PMT$ positive (the payment received). $N$ can be any value, because the balance is the same after every period. For $200 000 at $4.8\%$ per annum compounding monthly: $I(\%) = 4.8$, $PV = -200\,000$, $FV = 200\,000$, $PpY = CpY = 12$ gives $PMT = 800$.

For most exam questions, though, the relationship $d = \frac{r}{100}P$ is quicker than the solver.

Perpetuities and interest-only loans

The same condition appears in loans. If a borrower's repayment exactly equals the interest charged, the loan balance never falls: an interest-only loan. That is what 2025 Examination 2 Question 9a described: a $50 000 loan with $75 weekly repayments and a constant balance must have a weekly rate of $\frac{75}{50\,000} = 0.15\%$.