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VCE General Maths Unit 3 AOS 2
The Foundation: Modelling with Recurrence Relations
A recurrence relation is a rule that defines how to get from one term in a sequence to the next. This step-by-step approach is fundamental to modelling how financial values change over time. Every financial scenario in this topic is an application of one of three core recurrence models.
Arithmetic Sequences (Linear Change)
Describes a constant amount being added or subtracted each period. The recurrence relation is Vn+1 = Vn + d, where ‘d’ is the common difference.
Key Applications: Simple Interest, Flat Rate Depreciation.
Geometric Sequences (Proportional Change)
Describes the value being multiplied by a constant ratio each period. The recurrence relation is Vn+1 = R * Vn, where ‘R’ is the common ratio.
Key Applications: Compound Interest, Reducing Balance Depreciation.
Combined Arithmetic & Geometric
Models situations with both a percentage change and a fixed change. The recurrence relation is Vn+1 = R * Vn + d.
Key Applications: Reducing Balance Loans, Annuities.
Valuing Assets Over Time: Depreciation
Depreciation is the process by which an asset loses value. The value of an asset at any time is its book value. Understanding the different models is key to financial reporting and decision making.
Flat Rate (Prime Cost) Depreciation
The asset depreciates by a fixed amount each year, based on a percentage of its initial purchase price. This is a model of linear decay.
- Recurrence Rule:
Vn+1 = Vn - d - Nth Term Rule:
Vn = V0 - n*d
Reducing Balance Depreciation
The asset depreciates by a fixed percentage of its book value from the previous year. The amount of depreciation is highest at the start. This is a model of exponential decay.
- Recurrence Rule:
Vn+1 = R * Vn - Nth Term Rule:
Vn = R^n * V0
Unit Cost Depreciation
The asset depreciates based on usage (e.g., per kilometre driven) rather than time. This is another model of linear decay, where depreciation is calculated as `cost per unit * units used`.
The Dynamics of Interest: Loans and Investments
The principles of recurrence relations extend directly to modelling how money grows with interest in investments or decreases through repayments on a loan. The key is understanding compound interest and amortisation.
Compound Interest
Interest is calculated on the current balance of the account, meaning interest is earned on previously earned interest. This leads to exponential growth, modelled by a geometric sequence.
Reducing Balance Loans (Amortisation)
A loan where regular payments are made. Part of each payment covers interest, and the remainder reduces the outstanding principal. As the principal reduces, the interest charged also reduces. This is modelled with the combined recurrence relation Vn+1 = R*Vn - D.
Nominal vs. Effective Interest Rates
The nominal rate is the advertised annual rate. The effective rate is the true annual rate when compounding is considered. Use the `eff()` function on your CAS to compare financial products with different compounding periods fairly.
Structured Financial Products: Annuities & Perpetuities
Annuities and perpetuities are financial products designed to provide or manage streams of regular payments, essential for long-term planning like retirement.
Annuities (Providing an Income)
An investment that provides a regular stream of income for a set period from a lump sum. The balance gradually decreases. The model is identical to a reducing balance loan: Vn+1 = R*Vn - D, where D is the withdrawal.
Annuity Investments (Building Savings)
A savings account where regular deposits are made over time. The balance grows with both deposits and interest. The model is Vn+1 = R*Vn + D, where D is the deposit.
Perpetuities
An annuity designed to last forever. This is achieved when the regular withdrawal is exactly equal to the interest earned, so the principal never changes. The formula is D = V0 * r, where D is the payment, V0 is the principal, and r is the interest rate per period.
Essential Toolkit: Mastering Your CAS Calculator
While understanding the theory is crucial, most VCE problems are solved efficiently using the Time-Value of Money (TVM) Finance Solver on your CAS calculator. Proficiency with this tool is essential.
The TVM Finance Solver: Variable Guide
The most critical part of using the solver is the sign convention. Money that flows away from you (payments, investments) is NEGATIVE. Money that flows towards you (loans received, future value cashed out) is POSITIVE.
| Variable | Description | Common Sign Convention |
|---|---|---|
| N | Total number of payments. | – |
| I% | Nominal annual interest rate. | – |
| PV | Present Value (Principal). | Loan: +, Investment: – |
| PMT | Regular Payment amount. | Loan Repayment: –, Investment Deposit: – |
| FV | Future Value (Balance). | Loan Paid Off: 0, Investment Value: + |
| P/Y | Payments per Year. | Usually 12 (monthly) or 4 (quarterly). |
| C/Y | Compounds per Year. | Usually same as P/Y. |
Depreciation Explorer
Visually compare Flat Rate vs. Reducing Balance depreciation. Adjust the sliders to see how the book value of an asset changes over time under each model.
$150,000
12%
20%
10 years
Loan & Investment Calculator
A replica of the CAS Finance Solver. Fill in the known values, then click “Calc” for the variable you want to find. Remember the sign convention!
Results
Your calculated value will appear here.
Amortisation Table Generator
Enter loan details to generate a table that breaks down each payment into interest and principal reduction, showing exactly how a loan is paid off over time.
| Pmt (n) | Interest Paid | Principal Reduction | Balance |
|---|---|---|---|
| Your amortisation table will appear here. | |||
