Apply simple interest, compound interest and depreciation models to financial calculations, including future value, present value and effective annual rate

What this dot point is asking

QCAA wants you to apply the standard financial formulas (simple interest, compound interest and depreciation), compute future and present values, find an effective annual rate, and interpret the results in everyday contexts such as savings, loans and asset values. Financial mathematics is a recurring modelling context, and the key skills are choosing the right model and keeping rates and time periods consistent.

Simple interest

Simple interest is calculated only on the original principal, never on accumulated interest:

where $P$ is the principal, $r$ is the annual rate as a decimal, and $t$ is the time in years. Because the interest each period is constant, the balance grows arithmetically.

Compound interest

Compound interest adds the interest to the balance at the end of each compounding period, so later interest is earned on earlier interest:

where $n$ is the number of compounding periods per year ($n = 1$ annually, $2$ half-yearly, $4$ quarterly, $12$ monthly, $365$ daily). The balance grows geometrically with common ratio $1 + \tfrac{r}{n}$ per period. For continuous compounding the limit is $A = Pe^{rt}$.

Effective annual rate

The same nominal annual rate produces different real returns depending on how often it compounds. The effective annual rate (EAR) is the single annual rate that gives the same growth:

For a nominal $5\%$ compounded monthly, $\text{EAR} = (1 + 0.05/12)^{12} - 1 \approx 0.05116$, that is about $5.12\%$. The EAR lets you compare products quoted at different compounding frequencies on a fair basis.

Present value

Present value answers "how much must I invest now to reach a target amount $A$?" by rearranging the compound-interest formula:

This discounts a future amount back to today and underlies lump-sum settlements and savings goals. Because the discount factor $(1 + r/n)^{-nt}$ shrinks as the rate or the time grows, money promised further in the future is worth less now, the core idea of the time value of money.

Depreciation

Assets lose value over time, and two models are standard. Straight-line depreciation loses a constant dollar amount each year, an arithmetic model:

where $d$ is the fixed annual loss. Declining-balance depreciation loses a constant percentage of the current value each year, a geometric model:

Declining balance falls quickly at first and then more slowly, which often matches real assets such as vehicles and machinery.

In one sentence

Simple interest is $I = Prt$ on the principal only, compound interest is $A = P(1 + r/n)^{nt}$ with effective annual rate $(1 + r/n)^n - 1$, present value is $A(1 + r/n)^{-nt}$, and depreciation is either straight-line (arithmetic) or declining-balance (geometric, $V_t = P(1 - r)^t$).