QCE General Mathematics - Unit 4 - Loans, investments and annuities 1

Present Value of Ordinary Annuities | QCE General Mathematics

Learn reducing-balance loans, present value annuity formulae and repayment interpretation for QCE General Mathematics.

Updated 2026-05-18 - 4 min read

The present value of an ordinary annuity is used for loans or pensions where regular payments are made and interest is calculated before each payment. A reducing-balance loan is the main school example: the balance grows by interest, then falls when a repayment is made.

Recurrence model

For a reducing-balance loan:

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

where $r=1+i$, $i$ is the interest rate per period and $d$ is the regular payment.

The recurrence is useful because it shows the order of events: interest first, payment second.

Present value formula

The formula is:

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

where $A_{PV}$ is the loan amount or present value, $d$ is the periodic payment, $i$ is the interest rate per period and $n$ is the number of payments.

Worked example

Total interest

Total interest paid is:

$ \text{total payments}-\text{amount borrowed} $

If the total of repayments is $620\times60=37200$, then interest is about $37200-31212.35=5987.65$.

Reducing-balance loan tables

A reducing-balance loan repeatedly applies interest and then subtracts a repayment. For:

$ A_{n+1}=1.01A_n-500 $

the loan balance increases by $1\%$ per period before a $500$ repayment.

| Period | Calculation | Balance | |---:|---|---:| | $0$ | initial loan | $12000.00$ | | $1$ | $1.01(12000)-500$ | $11620.00$ | | $2$ | $1.01(11620)-500$ | $11236.20$ | | $3$ | $1.01(11236.20)-500$ | $10848.56$ |

This table is useful when the question asks for a balance after a small number of payments or when repayments/rates change.

Varying the repayment or interest

If the repayment increases, the loan is paid down faster. If the interest rate increases, more of each repayment is consumed by interest. A repayment that is too small may not reduce the loan at all.

Present value and repayments

The present value formula is best when the payment is constant and the number of payments is known. It can be rearranged with technology to find a repayment amount, loan size or number of payments. Always interpret total interest:

$ \text{total interest}=\text{total repayments}-\text{amount borrowed} $

Depth: ordinary annuity loan recurrence

A reducing-balance loan with regular repayments is usually modelled by:

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

where $r=1+i$ and $d$ is the repayment made after interest is added for the period. This order matters.

Present value formula

For an ordinary annuity:

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

This gives the present value of $n$ future payments of $d$ when the periodic interest rate is $i$.

Finding the repayment

Rearranging gives:

$ d=\frac{PV\cdot i}{1-(1+i)^{-n}} $

Final payment and overpayment

In recurrence tables, the final regular repayment may be larger than the remaining balance after interest. In real loans, the final payment is adjusted so the balance becomes exactly zero. If a question asks for the final smaller payment, calculate the balance just before the last payment and pay that amount.

Depth: total interest paid

Loan questions often ask for the total interest, not just the repayment. Once the regular repayment is known:

$ \text{total paid}=dn $

and:

$ \text{interest paid}=\text{total paid}-\text{amount borrowed} $

This interpretation is often the final mark in a finance question.

If repayments are rounded to the nearest cent, the final balance in a recurrence table may be a few cents above or below zero. That is a rounding effect, not a modelling error, unless the discrepancy is large.

Quick check

Sources

• QCAA General Mathematics subject page
• QCAA General Mathematics 2025 syllabus
• QCAA General Mathematics 2025 formula book
• ASIC Moneysmart: Loans