Annuities

An annuity is a series of equal payments made at regular intervals, with compound interest. This focus area was worth 4 marks in NESA's sample HSC paper, and annuities averaged about 7 marks a year in the 2020–2025 HSC papers. In the 2024 syllabus you use tables of interest factors rather than formulas, so the main skills are reading the right value from a table and knowing whether to multiply or divide.

• Identify an annuity: regular equal contributions into an account (saving), or regular equal withdrawals or repayments from a single sum (a loan or a retirement fund).
• Model an annuity in a table for up to four periods, and with a spreadsheet.
• Use a table of future value interest factors and a table of present value interest factors.
• Find the regular contribution needed to reach a goal, or the single sum needed now.
• Examine how the deposit, the rate and the time change the result, and use annuities to make decisions about loans and savings.

There are two kinds of annuity problem:

• Future value (saving): you pay in a fixed amount at the end of each period and want to know how much you will have at the end. Example: $200 a month into a savings account for 4 years, or regular contributions to superannuation.
• Present value (spending or borrowing): a single sum now pays out or is paid back by equal amounts each period. Example: a car loan repaid by equal monthly repayments, or a retirement fund (such as superannuation) that pays $3000 a month.

In this course, payments are made at the end of each period, and interest is compounded at the same time. Just as with compound interest, change the rate and time to match the period: for monthly payments, $r$ = annual rate ÷ 12 and $n$ = years × 12.

Each period: add the interest on the balance, then add the new contribution.

Worked example: three yearly deposits

Sam deposits $2000 at the end of each year into an account earning 5% p.a., compounded annually. Find the balance after 3 years.

Year 1 earns no interest because the first deposit only arrives at the end of the year. After 3 years Sam has $6305: $6000 of deposits and $305 of interest.

A spreadsheet does the same thing row by row: each row's end balance is the start balance × 1.05 + 2000, and filling the formula down gives as many years as you need.

A future value interest factor is the future value of an annuity of $1 per period. For any other contribution, multiply:

Part of a future value table (factors for an annuity of $1 per period):

Worked example: quarterly saving

Ella deposits $900 at the end of every quarter for 5 years into an account paying 6% p.a., compounded quarterly. Find the future value and the interest earned.

Step 1: rate and periods per quarter.

Step 2: read the factor in row 20, column 1.5%: 23.1237.

Step 3: multiply.

Step 4: interest = future value minus total deposits.

Ella will have $20 811.33, of which $2811.33 is interest.

Finding the contribution needed

If you know the goal, divide by the factor. To save $50 000 in 10 years at 6% p.a. compounded monthly: $r=0.5\%$ and $n=120$, so the factor is 163.8793.

About $305.10 a month is needed (to the nearest cent).

A present value interest factor is the single amount needed now to fund payments of $1 per period. It is used for loans and for funds that pay out regularly.

Part of a present value table:

For a loan, the present value is the amount borrowed. If you can afford repayments of $360 a month for 4 years at 6% p.a. compounded monthly, the most you can borrow is $360\times 42.5803$, which is $15 328.91.

Worked example: car loan repayments

Josh borrows $25 000 for a car at 7.2% p.a., compounded monthly, over 5 years. Find the monthly repayment, the total he repays and the total interest.

Step 1: $r=7.2\%÷12=0.6\%$ and $n=5\times 12=60$. The PV factor is 50.2621.

Step 2: the amount borrowed is the present value, so divide.

Step 3: totals.

Josh repays $497.39 a month, $29 843.40 in total, of which $4843.40 is interest.

Ask: "Is the big amount at the end (savings) or at the start (a loan or a fund)?" End means future value; start means present value.

Using a spreadsheet or the tables, you can see that:

• A higher rate or a longer time gives a much larger future value, because interest is earned on interest. Across the 1.5% column, tripling the time from 20 to 60 periods more than quadruples the factor.
• Bigger contributions increase the future value in proportion: double the deposit, double the total.
• For a loan, a longer term means smaller repayments but much more interest in total.
• Starting with an amount already in the account adds its compound growth, $PV(1+r{)}^n$, to the annuity's future value.

• Using the annual rate and the number of years when payments are monthly or quarterly.
• Reading from the FV table for a loan question. Loans use the PV table.
• Multiplying when you should divide (or the reverse). Check: a loan repayment must be much smaller than the amount borrowed.
• Forgetting to subtract total deposits (or the amount borrowed) when the question asks for the interest.
• Treating the factor as a percentage. It is a multiplier for $1 per period.

• Show the rate per period and the number of periods before you read the table; that line often earns a mark by itself.
• State the factor you used, then the multiplication or division, then the answer to the nearest cent.
• Compare questions (save more each month, or for longer; a 3-year or a 5-year loan) need a value for each option and a conclusion in words.
• If a formula is printed in the question, use it; otherwise use the table given.

• Investment and loans: compound interest and reducing-balance loans are the same ideas, one payment at a time.
• Managing money (Year 11): budgets decide how much can be saved or repaid each period.
• Algebraic relationships: annuity totals grow exponentially over time.