 Course

# VCE Further Mathematics Units 3 and 4 – Recursion and Financial Mathematics

2.1 Recurrence Relations
2.2 Interest Rates
2.3 Borrowing Money
2.4 Annuity

26 Lessons

This topic covers the use of first-order linear recurrence relations and technology to model and analyse a range of financial situations, and solve related problems involving interest, appreciation and depreciation, loans, annuities and perpetuities.

Depreciation of assets, including:

• review of the use of a first-order linear recurrence relation to generate the terms of a sequence
• use of a recurrence relation to model and compare (numerically and graphically) flat rate, unit cost and reducing balance depreciation of the value of an asset with time, including the use of a recurrence relation to determine the depreciating value of an asset after n depreciation periods, including from first principles for $n \le 5$
• use of the rules for the future value of an asset after n depreciation periods for flat rate, unit cost and reducing balance depreciation and their application.

Compound interest investments and loans, including:

• review of the concepts of simple and compound interest
• use of a recurrence relation to model and analyse (numerically and graphically) a compound interest investment or loan, including the use of a recurrence relation to determine the value of the compound interest loan or investment after $n$ compounding periods, including from first principles for $n \le 5$
• difference between nominal and effective interest rates and the use of effective interest rates to compare investment returns and the cost of loans when interest is paid or charged, for example, daily, monthly, quarterly
• rule for the future value of a compound interest investment or loan after n compounding periods and its use to solve practical problems.

Reducing balance loans (compound interest loans with periodic repayments), including:

• use of a first-order linear recurrence relation to model and analyse (numerically and graphically) the amortisation of a reducing balance loan, including the use of a recurrence relation to determine the value of the loan or investment after n payments, including from first principles for $n \le 5$
• use of a table to investigate and analyse the amortisation of a reducing balance loan on a step-by-step basis, the payment made, the amount of interest paid, the reduction in the principal and the balance of the loan
• use of technology with financial modelling functionality to solve problems involving reducing balance loans, such as repaying a personal loan or a mortgage, including the impact of a change in interest rate on repayment amount, time to repay the loan, total interest paid and the total cost of the loan.

Annuities and perpetuities (compound interest investments with periodic payments made from the investment), including:

• use of a first-order linear recurrence relation to model and analyse (numerically and graphically) the amortisation of an annuity, including the use of a recurrence relation to determine the value of the annuity after n payments, including from first principles for $n \le 5$
• use of a table to investigate and analyse the amortisation of an annuity on a step-by-step basis, the payment made, the interest earned, the reduction in the principal and the balance of the annuity
• use of technology to solve problems involving annuities including determining the amount to be invested in an annuity to provide a regular income paid, for example, monthly, quarterly
• simple perpetuity as a special case of an annuity that lasts indefinitely.

Compound interest investment with periodic and equal additions to the principal (an annuity investment), including:

• use of a first-order linear recurrence relation to model and analyse (numerically and graphically) annuity investment, including the use of a recurrence relation to determining the value of the investment after n payments have been made, including from first principles for $n \le 5$
• use of a table to investigate and analyse the growth of an annuity investment on a step-by-step basis after each payment is made, the payment made, the interest earned and the balance of the investment
• use of technology with financial modelling functionality to solve problems involving annuity investments, including determining the future value of an investment after a number of compounding periods, the number of compounding periods for the investment to exceed a given value and the interest rate or payment amount needed for an investment to exceed a given value in a given time.

source – VCE Mathematics Study Design 