An annuity is a sequence of equal payments made at regular intervals. The recurrence relation for an annuity with interest rate $r$ per period and payment $d$:
$$A_{n+1} = A_n(1 + r) - d \quad \text{(reducing balance / withdrawals)}$$
$$A_{n+1} = A_n(1 + r) + d \quad \text{(savings / contributions)}$$
In Unit 4, annuities may involve:
- Non-annual compounding (e.g., monthly repayments on a yearly rate)
- Changing payment amounts at a certain point
- Finding $n$ (time to reach a target balance)
- Mixed annuities: savings phase followed by drawdown phase
When the compounding period differs from the payment period, convert to an effective periodic rate:
$$r_{\text{effective}} = \left(1 + \frac{r_{\text{annual}}}{k}\right)^k - 1$$
where $k$ = number of compounding periods per year.
Example: 6% p.a. compounding monthly → monthly rate $= 6\%/12 = 0.5\%$ per month.
A perpetuity is an annuity that continues indefinitely. The present value of a perpetuity paying $d$ per period at rate $r$ per period is:
$$PV = \frac{d}{r}$$
This formula arises because $A_{n+1} = A_n(1+r) - d$ has a fixed point when $A_{n+1} = A_n$:
$$A = A(1+r) - d \implies d = Ar \implies A = \frac{d}{r}$$
A charity needs to pay \$12,000 per year indefinitely. An account earns 4% p.a.
$$PV = \frac{12000}{0.04} = \$300{,}000$$
A lump sum of \$300,000 invested at 4% p.a. will fund the payments forever.
A loan of \$40,000 is taken at 5% p.a. compounding monthly. After 2 years, the borrower increases monthly repayments by \$200.
Step 1: Find balance after 2 years (24 months) using recurrence with original payment.
Step 2: Use the new (higher) payment in the recurrence to find time to full repayment.
CAS is essential for iterating the recurrence relation over many periods.
APPLICATION: Perpetuities are used to fund scholarships, bonds, and endowments. Understanding present value helps evaluate whether a proposed fund is adequate for the intended purpose.
EXAM TIP: Always state the recurrence relation with the correct sign convention for the payment term ($-d$ for drawdown, $+d$ for savings). Confirm whether payments are made at the start or end of each period.
