QCE General Mathematics - Unit 4 - Loans, investments and annuities 2

Perpetuities and Future Value of Ordinary Annuities | QCE General Mathematics

Learn future value annuities, perpetuities and periodic-payment finance models for QCE General Mathematics.

Updated 2026-05-18 - 4 min read

Future value annuities model regular deposits into an investment. Perpetuities model payments that continue indefinitely. Both are ordinary annuity models when interest is applied before the regular payment.

Future value recurrence

For a savings annuity:

$ A_{n+1}=rA_n+d $

where the balance earns interest, then the regular deposit $d$ is added.

Future value formula

$ A_{FV}=d\left(\frac{(1+i)^n-1}{i}\right) $

This gives the accumulated value of regular payments after $n$ periods.

Perpetuities

A perpetuity has payments that continue forever. The formula is:

$ A=\frac{d}{i} $

where $d$ is the regular payment and $i$ is the interest rate per period.

Worked example

Perpetuity example

If a scholarship pays $2400$ per year from interest earned at $6\%$ p.a., then:

$ A=\frac{2400}{0.06}=40000 $

A fund of $40000$ is needed, assuming the interest rate remains fixed.

Future value with an existing balance

The standard future value annuity formula only accounts for the regular deposits. If there is already money in the account, compound the starting balance separately:

$ \text{final balance}=P(1+i)^n+d\left(\frac{(1+i)^n-1}{i}\right) $

Interpreting perpetuities

A perpetuity works because the interest earned each period exactly funds the payment:

$ d=Ai $

so:

$ A=\frac{d}{i} $

If the fund earns less than expected, the payment may no longer be sustainable. That is why perpetuity questions usually assume a fixed interest rate.

Depth: ordinary annuity timing

An ordinary annuity assumes the payment is made at the end of each period. For a savings plan, that means interest is earned first and then the deposit is added. For a loan repayment, interest is charged first and then the repayment is deducted.

Timing changes the answer. A deposit made at the beginning of each period has one extra period to earn interest, which is called an annuity due. The standard QCE formula book ordinary annuity formulas use end-of-period payments unless the question specifies otherwise.

Future value table method

The recurrence method is useful for checking formula answers.

The formula gives the same result:

$ 300\left(\frac{1.01^3-1}{0.01}\right)=909.03 $

Perpetuity interpretation

For a perpetuity, the capital is not being spent down. The interest earned each period funds the payment. If the fund is $A$ and the periodic rate is $i$, the interest per period is $Ai$. Setting this equal to the payment $d$ gives:

$ Ai=d $

so:

$ A=\frac{d}{i} $

If the payment is quarterly, use the quarterly rate. If the payment is monthly, use the monthly rate. The payment period and interest period must match.

Comparing annuity types

| Model | What is known | What is found | |---|---|---| | future value annuity | regular deposits, rate, number of periods | accumulated savings | | present value annuity | regular repayments, rate, number of periods | loan size or current value | | perpetuity | regular payment and rate | capital needed forever |

This table is often enough to decide which formula to use before substituting numbers.

Depth: sinking fund questions

A sinking fund is a savings plan designed to reach a future target. The future value annuity formula can be rearranged to find the regular deposit:

$ d=\frac{A_{FV}i}{(1+i)^n-1} $

Perpetuity answers should state the capital required now, not the total of all future payments. The total paid over an infinite time horizon is not finite, but the present capital can be finite because interest funds each payment.

Quick check

Sources

• QCAA General Mathematics subject page
• QCAA General Mathematics 2025 syllabus
• QCAA General Mathematics 2025 formula book
• ASIC Moneysmart: Saving and investing