Investment

Investment is about how money grows. Questions are usually short calculations with a formula from the reference sheet, plus a comparison or a recommendation.

• Simple interest: $I=Prn$, calculated in a variety of contexts and compared for different rates and times.
• Compound interest: $FV=PV(1+r{)}^n$, used to find the future value, the present value or the interest rate.
• Compounding periods: yearly, half-yearly, quarterly, monthly and daily.
• Comparing simple and compound interest, numerically or from a graph.
• Spreadsheets to model investments.
• Investment strategies: comparing savings accounts, shares and property.

Simple interest is paid only on the original amount invested, the principal. The interest is the same every period.

• $I$ = interest ($)
• $P$ = principal ($)
• $r$ = interest rate per period, as a decimal (3.5% p.a. is 0.035 per year)
• $n$ = number of periods

$r$ and $n$ must use the same time unit. "p.a." means per annum (per year), so with a yearly rate, $n$ must be in years: 9 months is $\frac{9}{12}=0.75$ years.

Worked example: simple interest for part of a year

$6500 is invested at 2.8% p.a. simple interest for 9 months. Find the interest and the final value.

The interest is $136.50, so the investment is worth $6500+136.50=6636.50$ dollars.

You can also rearrange $I=Prn$. To find the rate, $r=\frac{I}{Pn}$; to find the time, $n=\frac{I}{Pr}$. Remember to change a decimal rate back to a percentage.

With compound interest, the interest each period is added to the balance, and the next period's interest is calculated on the new, bigger balance. You earn interest on your interest.

• $FV$ = future value (the final balance)
• $PV$ = present value (the amount invested now)
• $r$ = interest rate per compounding period, as a decimal
• $n$ = number of compounding periods

The interest earned is $FV-PV$.

Compounding periods

Divide the yearly rate by the number of periods in a year, and multiply the years by the same number.

The more often interest is compounded, the slightly bigger the future value.

Worked example: future value, compounded monthly

$3000 is invested at 4.2% p.a., compounded monthly, for 5 years.

The investment grows to $3699.68, so the interest earned is $699.68.

Present value

The present value is how much must be invested now to reach a target amount later. Divide instead of multiply:

Example. How much must be invested now at 3.5% p.a., compounded annually, to have $12 000 in 6 years?

About $9762.01 is needed now. Check: the present value must be less than the future value.

Finding the interest rate

Substitute what you know and solve for $r$. You can take a root, or test sensible rates on the calculator until the future value matches.

Example. $6000 grows to $7400 in 5 years, compounded annually. Find the rate.

So $r\approx 0.0428$, about 4.3% p.a. Note that dividing the total growth (23.3%) by 5 would give 4.7%, which is wrong because it ignores compounding.

Appreciation and inflation

The same formula models anything that grows by a percentage each year. A house worth $850 000 that appreciates (grows in value) by 5% p.a. would be worth about $850000\times 1.05^3\approx 983981$ dollars after 3 years. Inflation works the same way: prices rising 3% a year means something costing $100 now would cost about $100\times 1.03^5\approx 115.93$ dollars in 5 years.

• Simple interest grows by the same amount each year: its graph is a straight line (linear growth).
• Compound interest grows by a bigger amount each year: its graph is a curve that gets steeper (exponential growth).
• For the same rate, both are equal after one year (with annual compounding), and compound interest is ahead after that. The gap keeps widening.

On a graph question, identify the straight line as simple interest and the upward curve as compound interest, and read values from the gridlines.

A spreadsheet can show the balance year by year:

Filling row 3 down repeats the calculation each year. The simple interest column adds the same $500 each time; the compound column multiplies by 1.05 each time. You may be asked what a formula does or what a cell will show.

A good answer to a "recommend" question links the choice to the person's time frame and goal. Money needed soon (a deposit in a year, an emergency fund) suits a savings account. Money not needed for many years can ride out falls in shares or property, so a higher-return investment may suit.

• Using a yearly rate with months (or a monthly rate with years) in $I=Prn$.
• Using the yearly rate with monthly compounding: divide $r$ by 12 and multiply the years by 12.
• Giving the future value when the question asks for the interest (or the other way round).
• Multiplying by $(1+r{)}^n$ when finding a present value: you must divide.
• Writing the rate as 5 instead of 0.05 in the formula.
• Rounding in the middle of a calculation: keep the full calculator value until the end.

• Write the formula, then show the substitution, then the answer with units ($) and rounding to the nearest cent.
• If a question says "compare" or "which is better", calculate both and finish with a sentence that names the better option and by how much.
• Check your answer is sensible: interest is usually much smaller than the principal; a present value is less than the future value.
• For investment strategy questions, mention at least two of return, risk and access, and connect them to the situation.

• Managing money (Year 11): percentage increase and decrease are the building blocks of interest.
• Algebraic relationships (Year 12): simple interest is a linear model and compound interest is an exponential model.
• Depreciation and loans (Year 12): declining-balance depreciation uses the same structure with $1-r$, and loans and credit cards charge compound interest.