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 standard financial formulas (simple interest, compound interest, depreciation) and to interpret the results in everyday contexts.

Simple interest

Interest is calculated only on the principal, not on accumulated interest.

$I = P r t$.

$A = P + I = P(1 + rt)$.

$P$ = principal, $r$ = annual rate (decimal), $t$ = time in years.

Compound interest

Interest is added to the principal at the end of each compounding period; subsequent interest is calculated on the new balance.

$A = P \left(1 + \dfrac{r}{n}\right)^{nt}$.

$n$ = compounding periods per year ($n = 1$ annually, $n = 2$ half-yearly, $n = 4$ quarterly, $n = 12$ monthly, $n = 365$ daily).

For continuous compounding, $A = P e^{rt}$.

Compounding frequency matters

The same nominal annual rate produces different actual returns under different compounding frequencies. The "effective annual rate" (EAR) is:

$\text{EAR} = \left(1 + \dfrac{r}{n}\right)^n - 1$.

A nominal $5$% compounded monthly has EAR $= 1.05^{12/12}$, no wait: $\text{EAR} = (1 + 0.05/12)^{12} - 1 = 5.1162$%.

Present value

The amount you must invest now to reach a future value:

$P = \dfrac{A}{(1 + r/n)^{nt}}$.

Used for retirement planning, lump-sum settlements.

Depreciation

Straight-line. Constant amount lost each year. Arithmetic sequence.

$V_t = P - dt$ where $d$ is the annual depreciation.

Declining-balance. Constant percentage lost each year. Geometric sequence.

$V_t = P(1 - r)^t$.

Worked example (loan repayment summary)

A \200,000$loan at$6$$30$years. The future value of a single$\$200\,000$ at the loan rate would be:

A = 200\,000 (1.005)^{360} = 200\,000 \cdot 6.023 = \1,204,515$.

(In practice, loan repayment calculations use the annuity formula, which appears later in QCE Maths Methods.)

Common traps

Mixing annual and monthly rates. If $r$ is annual, $n = 12$, and $t$ is in years. Do not also divide $t$.

Forgetting to subtract principal for interest. Total interest = $A - P$.

Using simple interest where compound is meant. Australian consumer products (savings, loans, credit cards) almost always compound.

Decimal vs percent. $5$% as $0.05$ in formulas. Using $5$ directly gives nonsense.

In one sentence

Simple interest is $I = Prt$ (interest on principal only); compound interest is $A = P(1 + r/n)^{nt}$ (interest compounded $n$ times per year) with effective annual rate $\text{EAR} = (1 + r/n)^n - 1$; present value is $P = A(1 + r/n)^{-nt}$, and depreciation is either straight-line (arithmetic) or declining-balance (geometric).