Investment and Loans

Investment and loans is one of the biggest focus areas in Standard 2: about 9 to 10 marks every year, in both multiple choice and longer written questions. The good news is that it uses only a handful of formulas, all on the reference sheet. Most marks are for choosing the right formula, getting $r$ and $n$ right, and setting out your working.

• Simple interest $I=Prn$, and comparing simple interest investments.
• Compound interest $FV=PV(1+r{)}^n$: finding future value, present value or the rate; the effect of the rate, term and compounding period; inflation and appreciation.
• Simple interest graphs are linear; compound interest graphs are exponential.
• Shares: reading share price graphs, dividends, dividend yield and brokerage costs.
• Depreciation: straight-line $S=V_0-Dn$ and declining balance $S=V_0(1-r{)}^n$.
• Reducing-balance loans: tables for up to four periods, spreadsheets, and how extra repayments change the loan.
• Credit cards: fees, interest-free periods, minimum payments and daily compound interest.

Simple interest is paid only on the original amount (the principal), so the interest is the same every period.

$P$ is the principal, $r$ the interest rate per period as a decimal, and $n$ the number of periods. For example, $4000 at 3.5% p.a. for 30 months: $n=2.5$ years, so $I=4000\times 0.035\times 2.5=350$ which is $350.

Compound interest is paid on the principal and on the interest already earned, so the balance grows faster and faster.

$FV$ is the future value, $PV$ the present value (the amount now), $r$ the rate per compounding period and $n$ the number of compounding periods.

Getting r and n right

Always change the annual rate and the time to match the compounding period:

Worked example: compound interest

$8000 is invested at 4.2% p.a., compounded monthly, for 5 years. Find the future value and the interest earned.

The future value is $9865.81, so the interest is $1865.81. With simple interest at the same rate, the interest would be $8000\times 0.042\times 5=1680$ which is $1680. Compounding earns more, and the gap grows the longer the money is invested.

Present value

To find how much to invest now, rearrange: $PV=\frac{FV}{(1+r{)}^n}$ To have $15 000 in 4 years at 3.9% p.a. compounded quarterly:

You would need to invest $12 843.09 now.

Inflation and appreciation use the same formula: a price that rises with inflation of 3% p.a. is multiplied by $1.03$ each year.

A share is a small part of a company. Shareholders may receive a dividend, a payment per share from the company's profit. Buying and selling shares usually costs brokerage, a fee paid to the broker (a flat amount or a percentage).

Here the dividend is the amount paid per share each year.

Example. A share costs $8.40 and pays a dividend of 36 cents a year. The dividend yield is $\frac{0.36}{8.40}\times 100=4.29\%$ (2 d.p.). Buying 250 of these shares costs $250\times 8.40=2100$ dollars plus brokerage; if brokerage is 0.5%, the total is $2100\times 1.005=2110.50$ which is $2110.50.

Franked dividends are not part of the course. Share price graphs show the price over time; read the highest and lowest prices, and the change over a period.

Comparing investments

• Savings accounts: low risk and easy to access, but usually a low interest rate.
• Term deposits: a fixed rate for a fixed time; usually a bit higher than a savings account, but the money is locked away.
• Shares: can grow faster and pay dividends, but the price can fall, so the risk is higher.
• Property: can increase in value and earn rent, but needs a large amount of money and has costs such as stamp duty and maintenance.

A compare or recommend question wants the return of each option (as a number where you can) and a comment on risk, access to the money and time.

Depreciation is the loss of value of an item, such as a car or machine, over time. $V_0$ is the purchase price and $S$ the salvage value (value after $n$ periods).

• Straight-line: loses the same dollar amount $D$ each year. The graph is a straight line going down.
• Declining balance: loses the same percentage $r$ of its current value each year. The graph is an exponential decay curve: big losses at first, smaller later.

Worked example: comparing methods

A ute is bought for $36 000. Compare its value after 4 years using (a) straight-line depreciation of $4000 a year and (b) declining-balance depreciation at 15% p.a.

After 4 years the ute is worth $20 000 by straight-line and $18 792.23 by declining balance. A spreadsheet table shows that this changes: after 6 years the straight-line value ($12 000) is lower than the declining-balance value ($13 577.38). Straight-line can reach zero; declining balance never does.

Most loans, such as car loans and home loans, are reducing-balance loans: each period, interest is charged on the amount still owing, then a repayment is made. Model it in a table.

Worked example: a loan table

Priya borrows $6000 at 0.8% interest per month and repays $1000 at the end of each month. Complete a table for 3 months.

Month 1: $6000\times 0.008=48$, and $6000+48-1000=5048$. Month 2: $5048\times 0.008=40.38$ After 3 months she owes $3121.09. The interest falls each month because the balance falls, so more of each repayment pays off the loan itself. That is why a home loan balance drops slowly at first and faster later.

Changing a loan

• Extra repayments or more frequent repayments (fortnightly instead of monthly) reduce the balance sooner, so less interest is charged and the loan is paid off earlier.
• A higher interest rate means more interest, a longer term or bigger repayments.
• The total paid is repayment × number of repayments; total interest is total paid minus the amount borrowed.

A spreadsheet repeats the table row by row; questions may show part of one and ask you to fill a cell or say when the balance reaches zero.

A credit card is a reducing-balance loan that you choose how fast to repay. Card interest rates are much higher than home or personal loan rates.

• Interest-free period: no interest if the full balance is paid by the due date.
• Fees: annual fees, late payment fees, cash advance fees (cash advances usually attract interest from the day they are taken out).
• Minimum payment: the smallest amount you must pay by the due date, often 2% to 3% of the balance. Paying only the minimum means the debt takes years to clear and costs a lot of interest.

In HSC questions, credit card interest is usually compounded daily. Use $FV=PV(1+r{)}^n$ with $r$ = annual rate ÷ 365 and $n$ = the number of days.

Example. $1200 is owed for 25 days at 19.5% p.a., compounded daily.

The interest is $16.13. Count days carefully: 3 May to 28 May is 25 days, or 26 days if the question says to include both the purchase date and the payment date.

Choosing credit

For a short-term need that can be repaid within the interest-free period, a credit card can cost nothing. For a larger purchase repaid over a few years, a personal loan usually has a much lower rate than a credit card. Long-term goals like a home use a home loan, with the lowest rates of all.

• Using the annual rate with monthly periods. Divide the rate and multiply the years.
• Giving the future value when the question asks for the interest, or the reverse.
• Writing 15% depreciation as $(1-15{)}^n$ or $(0.15{)}^n$. It is $0.85^n$.
• In a loan table, subtracting the repayment before adding the interest, when the question says interest is charged first.
• Rounding in the middle of a compound interest calculation. Keep the full calculator value until the end.
• Forgetting brokerage when finding the total cost of, or profit from, shares.

• Write the formula, then the substitution, then the answer. The substitution line usually earns a mark even if you mistype on the calculator.
• Money answers go to the nearest cent unless the question says otherwise.
• Compare questions (two investments, two depreciation methods, two loans) need a number for each option and a sentence saying which is better and why.
• When a graph shows simple and compound interest, the straight line is simple interest and the curve is compound interest.
• Reading a credit card statement: find the closing balance, the minimum payment, the due date and the interest rate before you start.

• Annuities: regular deposits or repayments with compound interest, solved with tables of interest factors.
• Algebraic relationships: compound interest and declining balance are exponential models; simple interest and straight-line depreciation are linear.
• Managing money (Year 11): percentage increase and decrease, and ways of paying.
• Earning money (Year 11): income is the starting point for saving and borrowing.