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Annuities and Perpetuities

General Mathematics

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Area Of Study

Recursion and financial modelling

Unit

Unit 3: General Mathematics Unit 3

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Annuities and Perpetuities

General Mathematics
01 May 2026

Annuities and Perpetuities

What is an Annuity?

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.

Future Value and Present Value

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

Future Value of an Annuity (savings)

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.)

Worked Example: Investment Annuity

Problem: \$200 invested each month for 3 years at 6% p.a. compounded monthly. Find the future value.

  • Rate per month: \(r = \frac{6}{12} = 0.5\% = 0.005\)
  • \(R = 1.005\), \(d = 200\), \(n = 36\) months

Recurrence: \(V_{n+1} = 1.005 \times V_n + 200, \quad V_0 = 0\)

Using CAS/TVM solver: FV = \$7856.40 (approximately)

Worked Example: Drawdown Annuity

Problem: \$200,000 invested at 4.8% p.a. compounded monthly. Monthly withdrawal of \$1200.

  • Rate per month: \(r = \frac{4.8}{12} = 0.4\%\)
  • \(R = 1.004\), \(d = 1200\)

Recurrence: \(V_{n+1} = 1.004 \times V_n - 1200, \quad V_0 = 200000\)

  • If interest earned per period < \$1200, balance decreases
  • Interest at start: \(0.004 \times 200000 = \$800 < \$1200\) → balance decreases each month

Use CAS to find when \(V_n \leq 0\).

Perpetuities

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%
\(V_0 = \frac{500}{0.0025} = \$200,000\)

Finding Present Value

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.

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