Financial Maths
Apply mathematics to real financial decisions: depreciation, annuities, and present and future value calculations that underpin loans, superannuation, and investment.
Depreciation
Depreciation is the decrease in value of an asset over time. There are two common methods: straight-line (flat-rate) and reducing balance (compound rate).
Straight-Line Depreciation
The asset loses the same dollar amount each period.
Value = P − (D × n)
P = purchase price, D = annual depreciation, n = years
Reducing Balance Depreciation
The asset loses a fixed percentage each period.
Value = P × (1 − r)n
P = purchase price, r = rate, n = years
Worked Example: Reducing balance depreciation
A car is purchased for $30,000 and depreciates at 15% per year. What is its value after 4 years?
Value = 30,000 × (1 − 0.15)4 = 30,000 × (0.85)4
= 30,000 × 0.5220 ≈ $15,661
Future Value and Compound Interest
Future value (FV) is what an investment or loan will be worth at a future date, accounting for compound interest.
Future Value Formula
FV = PV × (1 + r)n
PV = present value, r = interest rate per period, n = number of periods
Worked Example: Compound interest
$5,000 is invested at 6% p.a. compounded monthly for 3 years. Find the future value.
Monthly rate: r = 0.06/12 = 0.005; n = 3 × 12 = 36 periods
FV = 5000 × (1.005)36 = 5000 × 1.1967 ≈ $5,983.40
Annuities: Present and Future Value
An annuity is a series of equal payments made at regular intervals. Superannuation contributions and loan repayments are examples of annuities.
Future Value of Annuity
FV = M × [(1+r)n − 1] / r
M = regular payment, r = rate/period, n = periods
Present Value of Annuity
PV = M × [1 − (1+r)−n] / r
Used to find the value of a loan today
Worked Example: Saving for a goal
Zara deposits $200 per month into an account earning 6% p.a. (compounded monthly). How much will she have after 2 years?
M = $200, r = 0.005, n = 24
FV = 200 × [(1.005)24 − 1] / 0.005
= 200 × [1.1272 − 1] / 0.005
= 200 × 25.432 ≈ $5,086.40
Key Vocabulary
Present Value (PV)
The current worth of a future sum of money, given a specified rate of return. Used to compare the value of money across time.
Future Value (FV)
The value of an investment or asset at a specific date in the future, based on assumed growth rate.
Annuity
A series of equal payments at regular intervals, such as monthly mortgage repayments or superannuation contributions.
Depreciation
The reduction in the value of an asset over time due to wear and obsolescence. Calculated using flat-rate or reducing balance methods.
Knowledge Check
Apply financial mathematics to real-world problems.
Question 1
A laptop costs $1,200 and depreciates by $150 per year (straight-line). What is its value after 5 years?
Question 2
$10,000 is invested at 5% p.a. compound interest for 3 years. What is the future value? (round to nearest dollar)
Question 3
A machine worth $50,000 depreciates at 20% p.a. (reducing balance). What is its value after 2 years?
Question 4
Which interest type gives greater returns on an investment over time?
Question 5
Liam deposits $500 per month for 12 months at 0.5% monthly interest. Using FV = M[(1+r)n−1]/r, what is the closest future value?
Key Concepts Summary
- ●Straight-line depreciation: V = P − D × n (constant annual reduction).
- ●Reducing balance depreciation: V = P × (1 − r)n (exponential decay model).
- ●Future value: FV = PV × (1 + r)n for a lump sum investment.
- ●Future value of annuity: FV = M × [(1+r)n − 1] / r for regular payments.
- ●Compound interest grows faster than simple interest due to earning interest on accumulated interest.