How do regular payments build up or draw down a fund over time?
Model annuities and annuity-investments with recurrence relations and find balances, payments and the time to exhaust or reach a target.
How to model an annuity that pays out and an annuity-investment that builds up, using recurrence relations, and find balances, payment sizes and how long a fund lasts or takes to reach a target.
Reviewed by: AI editorial process; not yet individually human-reviewed
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What this dot point is asking
You must set up the right recurrence, find balances, solve for the payment or the time, and recognise the difference between paying in and drawing out.
Two kinds of annuity
The same machinery models both saving and spending.
A draw-down annuity is structurally identical to a reducing-balance loan: both subtract a regular amount after interest. The difference is only in interpretation.
Building up: annuity-investments
When regular deposits earn compound interest, the balance grows faster than simple saving because earlier deposits earn interest for longer. Typical questions ask for the balance after a set time or the deposit needed to reach a target.
Drawing down: pensions
When a lump sum earns interest but regular withdrawals are taken, the fund falls. Questions ask how long it lasts or the largest sustainable withdrawal.
Solving for time or payment
To find how long a draw-down fund lasts, iterate until the balance hits zero. To find the deposit that reaches a target, the finance solver (or trial) finds the payment that lands the balance on the goal. State whether payments are at the start or end of each period, as this shifts the totals.