Use simple and compound interest formulas to find future values, present values, interest rates and time periods

What this dot point is asking

NESA wants you to apply the simple and compound interest formulas to investments and debts, switch between an annual interest rate and a per-period rate, and solve for any one of $A$, $P$, $r$ or $n$ given the others. You also need to compare scenarios with different compounding frequencies.

The answer

Simple interest

Simple interest is paid only on the original principal. After $n$ time periods at per-period rate $r$,

$P$ is the principal, $I$ is total interest, $A$ is the total amount. The graph of $A$ against $n$ is a straight line.

Compound interest

Compound interest is added to the principal at the end of each period and earns interest in subsequent periods. After $n$ compounding periods at per-period rate $r$,

The graph of $A$ against $n$ is exponential. For the same nominal rate and time, compound interest exceeds simple interest after the first period.

Per-period rate and number of periods

Rates are usually quoted as a nominal annual rate, but interest may compound more often than annually. Convert before applying the formula.

• Annual compounding: $r = \frac{R}{1}$, $n =$ years.
• Monthly: $r = \frac{R}{12}$, $n = 12 \times$ years.
• Quarterly: $r = \frac{R}{4}$, $n = 4 \times$ years.
• Daily: $r = \frac{R}{365}$, $n = 365 \times$ years.

Where $R$ is the nominal annual rate expressed as a decimal.

Present value

The present value $P$ is the amount you must invest today to grow to $A$ in $n$ periods. Rearranging,

Discounting a future amount back to the present is the same operation as compounding in reverse.

Solving for the rate or time

Solving $A = P(1 + r)^n$ for $r$ or $n$:

The time formula needs logarithms, which is fair game in Maths Advanced.

Effective annual rate

The effective annual rate makes different compounding frequencies comparable. If the nominal annual rate is $R$ compounded $m$ times per year,

A nominal $6\%$ compounded monthly is an effective $6.17\%$, slightly more than $6\%$ compounded annually.

Worked examples

Simple vs compound

Compare \1000$at$5%$per annum for$10$ years under simple and compound interest (annual compounding).

Simple: A = 1000(1 + 0.05 \cdot 10) = \1500$.

Compound: A = 1000(1.05)^{10} = 1000 \cdot 1.62889 \approx \1628.89$.

Compounding earns about \128.89$ more over the decade.

Quarterly compounding

\8000$at$4%$per annum compounded quarterly for$3$ years.

$r = \frac{0.04}{4} = 0.01$, $n = 4 \cdot 3 = 12$.

A = 8000 (1.01)^{12} = 8000 \cdot 1.12683 \approx \9014.62$.

Present value

You need \20000$in$5$years and can earn$6%$ per annum compounded annually. Invest now:

P = \frac{20000}{(1.06)^5} = \frac{20000}{1.33823} \approx \14945.16$.

Solving for time

How long does it take \2000$to double at$7%$ per annum compounded annually?

$2 = (1.07)^n \implies n = \frac{\ln 2}{\ln 1.07} \approx \frac{0.6931}{0.0677} \approx 10.24$ years.

So $11$ full years are required to reach or exceed \4000$.

Common traps

Using the annual rate with monthly periods. If interest compounds monthly, the rate in the formula is $R / 12$, not $R$. The number of periods is months, not years.

Mixing up simple and compound. A linear question uses $A = P(1 + r n)$. An exponential question uses $A = P(1 + r)^n$. Read the wording carefully.

Forgetting to discount. A present value question asks for the amount today, so divide by $(1 + r)^n$ (or multiply by $(1 + r)^{-n}$).

Mis-rounding intermediate steps. Carry the unrounded compound factor to the final calculation, then round to cents at the end. Rounding $(1.005)^{48}$ to $1.27$ instead of $1.27049$ shifts the answer by several dollars.

Confusing nominal and effective rates. A nominal $12\%$ compounded monthly is an effective $12.68\%$. The two are not interchangeable.

In one sentence

Compound interest grows a principal as $A = P(1 + r)^n$ where $r$ is the per-period rate and $n$ the number of periods, and the same formula rearranges to give present value, the required rate, or the number of periods to reach a target amount.