An annuity is a financial arrangement involving regular, equal payments combined with compound interest. There are two main types:
| Type | Description | Recurrence relation |
|---|---|---|
| Investment annuity (savings) | Regular deposits + interest earned | $V_{n+1} = R \cdot V_n + d$ |
| Annuity (pension/drawdown) | Regular withdrawals from invested funds | $V_{n+1} = R \cdot V_n - d$ |
Where $R = 1 + \frac{r}{100}$ and $d$ = payment per period.
| Term | Definition |
|---|---|
| Future value (FV) | The value of the annuity at the end of $n$ periods |
| Present value (PV) | The amount needed now to achieve a given future outcome |
If you invest $d$ per period at interest rate $r\%$ per period for $n$ periods:
$$FV = d \times \frac{(1+r)^n - 1}{r}$$
(In VCE, use technology/recurrence rather than memorising this formula.)
Problem: \$200 invested each month for 3 years at 6% p.a. compounded monthly. Find the future value.
Recurrence: $V_{n+1} = 1.005 \times V_n + 200, \quad V_0 = 0$
Using CAS/TVM solver: FV = \$7856.40 (approximately)
Problem: \$200,000 invested at 4.8% p.a. compounded monthly. Monthly withdrawal of \$1200.
Recurrence: $V_{n+1} = 1.004 \times V_n - 1200, \quad V_0 = 200000$
Use CAS to find when $V_n \leq 0$.
A perpetuity is a special annuity where the balance never decreases — the interest earned exactly equals the withdrawal.
$$\text{Perpetuity condition: } d = \frac{r}{100} \times V_0$$
Rearranged: $$V_0 = \frac{d \times 100}{r} = \frac{d}{r/100}$$
Example: To maintain a \$500/month payment indefinitely at 3% p.a. monthly compounding:
Monthly rate = 0.25%
$V_0 = \frac{500}{0.0025} = \$200,000$
Present value answers: “How much do I need now to fund a given plan?”
Use TVM solver with $FV = 0$ (loan fully paid) or solve the recurrence relation backwards.
KEY TAKEAWAY: An annuity combines compound interest with regular payments. A perpetuity is the special case where interest = withdrawal, keeping the balance constant forever.
EXAM TIP: For annuity problems, set up the recurrence relation first: identify $R$, $d$, and $V_0$. Then use CAS to find the required quantity. Always state whether $d$ is positive (deposit) or negative (withdrawal).
COMMON MISTAKE: Not converting the annual interest rate to the correct compounding period. Monthly compounding requires a monthly rate.
