What is wrong sign on the payment?

Repayments and withdrawals are subtracted; deposits are added. A sign slip turns a shrinking loan into a growing one.

QLDGeneral MathematicsSyllabus dot point

Topic 1: Loans, investments and annuities - how do we model financial situations with recurrence relations and amortisation?

Model compound interest investments, reducing-balance loans and annuities using first-order recurrence relations, compute future value, repayment and balance using technology, build and interpret an amortisation table, and analyse the effect of changing the rate, repayment or compounding period

A focused answer to the QCE General Mathematics Unit 4 dot point on loans, investments and annuities. Covers compound interest, reducing-balance loans, annuities and recurrence modelling, amortisation tables, and the effect of changing rate or repayment, with arithmetic-verified worked examples for IA3 and the external assessment.

Generated by Claude Opus 4.76 min answerUpdated 2026-05-29

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking

QCAA wants you to model real financial products with recurrence relations: a compound interest investment, a reducing-balance loan, and an annuity that pays out or pays down a balance. You compute balances and repayments with technology (a financial solver or recurrence on CAS), construct an amortisation table that splits each payment into interest and principal, and explain what happens when the interest rate, repayment size or compounding frequency changes. This is the financial core of Unit 4 Topic 1 and a heavy feature of IA3 and the external assessment.

The answer

The compounding recurrence

Every product in this topic is built from one recurrence: each period the balance is multiplied by a growth factor and then adjusted by a payment. Let $r$ be the interest rate per period as a decimal and $V_n$ the balance after $n$ periods.

V_{n+1} = V_n (1 + r) + D,

where $D$ is the payment added each period. The rate per period is the annual rate divided by the number of compounds per year. For an annual rate of $6$ percent compounding monthly, $r = 0.06 / 12 = 0.005$ .

Compound interest investments

With no regular payment ( $D = 0$ ) the recurrence becomes pure compound growth. The closed form is

V_n = V_0 (1 + r)^n,

where $V_0$ is the principal and $n$ is the number of compounding periods. Adding a regular deposit ( $D > 0$ ) makes it a savings annuity.

Reducing-balance loans

A loan starts as a debt. Each period interest is charged and a repayment $R$ is subtracted, so $D = -R$ :

V_{n+1} = V_n (1 + r) - R.

The loan is fully repaid when the balance reaches zero. Because the balance shrinks, the interest portion of each repayment falls and the principal portion rises over the life of the loan.

Annuities

An annuity is an account paying out equal regular withdrawals from an invested lump sum. It uses the same recurrence with $R$ as the withdrawal. The account is exhausted when the balance hits zero.

Amortisation tables

An amortisation table records, for each payment: the interest charged that period, the principal repaid (payment minus interest), and the new reduced balance.

\text{interest} = r \times \text{balance}, \qquad \text{principal repaid} = R - \text{interest}.

Effect of changing inputs

Higher interest rate. More interest per period, so total cost rises and the loan takes longer (or needs larger repayments) to clear.
Larger repayment. The loan clears faster and total interest paid falls.
More frequent compounding. More interest accrues over a year for the same nominal rate, increasing investment growth and loan cost.

Worked example

One row of a reducing-balance loan

A $\$ 20,000 $loan is charged interest at$ 9 $percent per year compounding monthly, with monthly repayments of$ \ $415$ .

Step 1: Monthly rate.

r = \frac{0.09}{12} = 0.0075.

Step 2: Interest for month 1.

0.0075 \times 20000 = 150.

Verify: $20000 \times 0.0075 = 150$ . Interest charged is $\$ 150$.

Step 3: Principal repaid in month 1.

415 - 150 = 265.

Step 4: Balance after month 1.

20000 - 265 = 19735,

or equivalently $20000 \times 1.0075 - 415 = 20150 - 415 = 19735$ .

Verify both ways: $20000 \times 1.0075 = 20150$ and $20150 - 415 = 19735$ . The new balance is $\$ 19,735$.

Step 5: Interest for month 2.

0.0075 \times 19735 = 148.0125 \approx 148.01.

The interest portion has fallen from $\$ 150 $to about$ \ $148.01$ , so more of the second repayment goes to principal. This is the defining feature of reducing-balance amortisation.

Common traps

Using the annual rate instead of the rate per period: For monthly compounding divide the annual rate by $12$ first. Using $0.09$ instead of $0.0075$ overstates interest twelvefold.
Wrong sign on the payment: Repayments and withdrawals are subtracted; deposits are added. A sign slip turns a shrinking loan into a growing one.
Computing interest on the original balance every period: That is simple interest. Reducing-balance interest is charged on the current balance, which falls each period.
Splitting the payment wrongly in the table: Interest comes first ( $r$ times balance), then principal is the remainder of the payment. Reversing the order corrupts the whole table.
Ignoring compounding frequency when comparing products: More frequent compounding raises the effective annual rate, so two products with the same nominal rate are not equivalent.

Exam-style practice questions

Practice questions written in the style of QCAA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

2021 QCAA4 marksDetermine the monthly repayment on a reducing balance loan of $720 000 at 4.8% p.a. over 25 years. Give your answer to the nearest dollar.

Show worked answer →

Step 1 - find the periodic rate and number of periods (1 mark). Monthly rate i = 4.8% / 12 = 0.4% = 0.004, and n = 25 x 12 = 300 months.

Step 2 - choose and substitute the annuity (present value) rule (1 mark). With principal A = 720 000, A = M x (1 - (1 + i)^(-n)) / i, so 720 000 = M x (1 - 1.004^(-300)) / 0.004.

Step 3 - solve for M (1 mark). The bracket evaluates to about 174.520, so M = 720 000 / 174.520 = 4125.578.

Step 4 - round and state (1 mark): the monthly repayment is about $4126.

Keep full calculator precision in the bracket and only round at the end; rounding 174.52 too early shifts the repayment by a few dollars.

2022 QCAA5 marksA couple borrow money to complete home renovations. Their bank has loaned the amount at 2.4% p.a. compounding monthly with repayments of $993.14 each month for 15 years. a) Determine the amount of money borrowed. [3 marks] b) Write a recurrence relation for the amount owing after n months. [2 marks]

Show worked answer →

a) Amount borrowed (3 marks). Set up the parameters (1 mark): i = 2.4% / 12 = 0.002, n = 15 x 12 = 180, M = 993.14. Substitute into the present-value annuity rule (1 mark): A = M x (1 - (1 + i)^(-n)) / i = 993.14 x (1 - 1.002^(-180)) / 0.002. Evaluate (1 mark): A = 150 000.29, so they borrowed about $150 000.

b) Recurrence relation (2 marks). A reducing-balance loan multiplies by (1 + i) each month and subtracts the repayment (1 mark for the correct form): A_(n+1) = (1 + i) A_n - R. Substituting i = 0.002 and R = 993.14 (1 mark): A_(n+1) = 1.002 A_n - 993.14, with A_0 = 150 000.

Part b) rewards both the general structure (multiply by 1 + i, subtract the repayment) and substituting the actual monthly rate and repayment.

2022 QCAA4 marksAn investment of

50 000 that compounds interest monthly is modelled by the recurrence relation A_(n+1) = 1.00375 A_n where A_0 = 50 000. a) What would be the advertised interest rate per annum, compounding monthly? [2 marks] b) How many months would it take for the value of the investment to exceed

51 000? [2 marks]

Show worked answer →

a) Advertised rate (2 marks). The multiplier is 1 + i where i is the monthly rate, so 1.00375 = 1 + i gives i = 0.00375 per month (1 mark). The advertised annual rate compounding monthly is 12 x 0.00375 = 0.045 = 4.5% p.a. (1 mark).

b) Months to exceed $51 000 (2 marks). Iterate the recurrence (1 mark): A_1 = 50 187.50, A_2 = 50 375.70, A_3 = 50 564.61, A_4 = 50 754.23, A_5 = 50 944.56, A_6 = 51 135.60. The balance first passes$ 51 000 at n = 6 (1 mark), so it takes 6 months.

A nominal annual rate compounding monthly is just the monthly rate times 12 (it is not compounded to get the annual figure here), and the count of months is the first n for which the balance strictly exceeds $51 000.