Module 7 · L4 of 12~25 minMST-12-S2-03⚡ +50 XP available
Annuities
You win the lottery and are offered a choice: $\$1$ million today, or $\$50,000$ every year for 30 years. Which is better? The answer depends on interest rates, and on a mathematical concept called an annuity. An annuity is a series of equal payments made at regular intervals. They appear everywhere: superannuation contributions, mortgage repayments, insurance premiums, and pension payments. Understanding annuities lets you value future cash flows in today's dollars, and today's dollars in future terms.
Today's hook, You deposit $\$200$ every month into an account earning 4.8% p.a. compounded monthly. After one year, is the total simply $\$200$ × 12 = $2,400? Predict before reading on.
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Orient to annuities
Connect the savings scenario to the lesson goals, formulas and essential language.
Worksheets
Practise this lesson
Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
You deposit $\$200$ every month into an account earning 4.8% p.a. compounded monthly. After one year, how much do you have? Is it simply $\$200$ × 12 = $2,400?
Before reading on write your gut feeling. We will revisit this at the end of the lesson.
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Beyond the syllabus. The annuity definition, the contribution table and the spreadsheet model are core. The closed-form FV/PV formulas and their geometric series explanation are an extension / alternate method, and annuity due timing is extension — the exam works with supplied interest-factor tables and end-of-period payments.
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Key ideas for this lesson
reference
An annuity converts a stream of equal payments into a single value, either in the future (FV) or in today's dollars (PV).
Future Value (FV): what a series of regular deposits grows to. Each deposit earns compound interest for a different number of periods.
Present Value (PV): what a stream of future payments is worth right now. Used to compare "lump sum today" vs "payments over time".
Variables: $M$ = payment per period, $r$ = interest rate per period (decimal), $n$ = total number of payments.
Annuities convert a stream of payments into a single lump sum (present or future), this lets you compare different financial options on equal terms.
Each payment earns different interest
In an ordinary annuity, the first payment earns interest for $n-1$ periods; the last earns no interest at all. The FV formula sums this geometric series.
Ordinary vs annuity due
Ordinary: payments at end of period (mortgages, loans). Annuity due: payments at start (rent, leases). FV_due = FV_ordinary × (1+r).
r is per period
If payments are monthly and the annual rate is 6%, use $r = 0.06/12 = 0.005$ per month. Always match the rate period to the payment period.
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What you will master
Know
Key facts
Definition of an annuity
Ordinary annuity vs annuity due
FV and PV formulas with variables M, r, n
Understand
Concepts
Why each payment earns a different amount of interest
The time value of money
When to use FV vs PV
Can do
Skills
Calculate FV of regular deposits
Calculate PV of regular payments
Identify real-world annuity situations
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Key terms
AnnuityA sequence of equal payments made at equal time intervals.
Ordinary annuityPayments made at the end of each period. Used for most loans and mortgage repayments.
Annuity duePayments made at the beginning of each period. Used for rent, leases, and insurance premiums.
Future value (FV)The total accumulated value of a series of payments at a future date, including all compound interest earned.
Present value (PV)The current lump-sum value equivalent to a series of future payments, discounted at the given interest rate.
Time value of moneyThe principle that a dollar today is worth more than a dollar in the future, because today's dollar can earn interest.
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Define an annuity
Identify ordinary annuities, annuities due and the variables used in each model.
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What is an annuity?
core concept
An annuity is a sequence of equal payments made at equal time intervals. The word comes from the Latin annus (year), but the payments can be monthly, fortnightly, quarterly, any regular period.
Real-world examples:
Monthly mortgage repayments, you pay a fixed amount to the bank each month.
Fortnightly superannuation contributions, employer deposits equal amounts each fortnight.
Annual insurance premiums, same amount every year.
Weekly rent payments, equal rent each week.
Monthly pension payments, equal income each month from retirement savings.
Key variables for annuity calculations:
$M$ = the payment amount per period
$r$ = the interest rate per period (as a decimal, always divide annual rate by periods per year)
$n$ = the total number of payments
Ordinary annuity vs annuity due: If payments occur at the end of each period, it's an ordinary annuity (most HSC problems). If payments occur at the beginning of each period, it's an annuity due, and each payment earns one extra period of interest: $FV_{\text{due}} = FV_{\text{ordinary}} \times (1+r)$.
An annuity is a sequence of equal payments at equal time intervals. Variables: M = payment per period, r = interest rate per period (annual rate ÷ periods per year), n = total number of payments.
Pause, copy the annuity definition (equal payments at equal intervals) and all three variable meanings: M = payment per period, r = interest rate per period, n = total number of payments into your book.
Quick check: Monthly mortgage repayments are an example of which type of annuity?
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Calculate future value
Use the future value formula and explain how regular payments accumulate interest.
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Future value, what will regular savings become?
core concept
An annuity is a series of equal payments M at equal intervals, with r as the per-period interest rate and n as the total number of payments. Each deposit earns compound interest for a different number of periods, the first earns interest for n − 1 periods, the last earns none. The FV formula adds all those compounding amounts into one result: FV = M × [(1 + r)ⁿ − 1] / r.
The future value of an ordinary annuity tells us how much a series of regular deposits will grow to by the end of the investment period.
$$FV = M \times \frac{(1+r)^n - 1}{r}$$
where $M$ = payment per period, $r$ = interest rate per period (decimal), $n$ = number of payments.
Why the formula works: The first payment is deposited one period before the end and earns compound interest for $n-1$ periods. The second earns interest for $n-2$ periods. The last payment earns no interest (it's just deposited). Summing these as a geometric series gives the formula above.
Worked example: $200 deposited monthly at 4.8% p.a. compounded monthly for 3 years.
$r = 0.048/12 = 0.004$ per month. $n = 36$.
$FV = 200 \times \dfrac{(1.004)^{36} - 1}{0.004} = 200 \times \dfrac{1.1547 - 1}{0.004} = 200 \times 38.68 = \$7{,}736$
Total deposited = $200 \times 36 = \$7{,}200$. Interest earned = $536.
Power of regularity: The longer the term, the more dramatically the interest compounds. Starting super contributions 10 years earlier can nearly double the final balance, small consistent deposits over long periods are extraordinarily powerful.
Future value of an ordinary annuity: FV = M × [(1 + r)^n − 1] / r. The first payment earns interest for n − 1 periods; the last earns none. Interest earned = FV − (M × n).
Pause, copy FV = M × [(1 + r)ⁿ − 1] / r with labels for each variable, and note that total interest earned = FV − (M × n) into your book.
True or false: In an ordinary annuity, the last payment deposited earns the most compound interest over the investment period.
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Model annuities period by period
Build matching table and spreadsheet models for an ordinary annuity.
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Building the annuity period by period, the table
core concept
The formula in card 06 gives you the answer in one line, and hides everything that produced it. Before trusting it, build the same annuity slowly, one period at a time. The syllabus asks you to model an annuity in tabular form for up to four periods, and a table of four rows is enough to see the whole mechanism.
Take $\$2000$ paid in at the end of each year, earning $5\%$ per annum, for $4$ years. Each row does the same three things: earn interest on what is already there, add the new contribution, carry the total forward.
Period
Opening balance
Interest at $5\%$
Contribution
Closing balance
1
$\$0.00$
$\$0.00$
$\$2000$
$\$2000.00$
2
$\$2000.00$
$\$100.00$
$\$2000$
$\$4100.00$
3
$\$4100.00$
$\$205.00$
$\$2000$
$\$6305.00$
4
$\$6305.00$
$\$315.25$
$\$2000$
$\$8620.25$
Period 1 earns nothing , and that is not a mistake in the table. In an ordinary annuity the contribution arrives at the end of the period, so during period 1 the account is empty and there is nothing for interest to act on. That single row is the definition from card 05, made visible.
The table and the formula are the same calculation. Card 06 gives $FV = 2000 \times \dfrac{(1.05)^4 - 1}{0.05} = 2000 \times 4.310125 = \$8620.25$ , which is the closing balance of period 4 to the cent. If your table and your formula disagree, one of them is wrong, and the table is the one you can check line by line.
Do not round as you go. Period 3's interest is $\$205.00$ exactly, but period 4's is $\$315.25$ . Rounding each interest figure to the nearest dollar as you fill the table walks the final balance away from the formula, and then you cannot tell a rounding drift from a real error.
To model an annuity in tabular form, use one row per period with columns for opening balance, interest, contribution and closing balance. Each closing balance becomes the next opening balance. In an ordinary annuity period 1 earns no interest, because the first contribution arrives at the end of that period. The final closing balance must equal the FV formula.
Pause, copy the five column headings, the rule that each closing balance becomes the next opening balance, and the reason period 1 earns no interest into your book.
Quick check: In the table above, where does the opening balance of period 3 come from?
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The same table in a spreadsheet
core concept
Four rows can be done by hand. Thirty years of monthly contributions is $360$ rows, and that is where the spreadsheet earns its place: you write the pattern once and fill it down.
Set out five columns for the same annuity, $\$2000$ a year at $5\%$ :
Cell
What goes in it
Fill down?
A2
the period number, 1
yes, =A2+1
B2
opening balance, 0
yes, =E2
C2
interest, =B2*0.05
yes
D2
contribution, 2000
yes
E2
closing balance, =B2+C2+D2
yes
One cell carries the whole model: B3 is =E2 . The new opening balance is the previous closing balance, and every row after it inherits that link when you fill down. Get that one reference right and the other four columns follow.
The check that catches a broken model. The closing balance in the period-4 row must read $\$8620.25$ , matching the formula in card 06 exactly. If it does not, look first at column B: a model that reads its opening balance from the wrong row, or from the contribution instead of the closing balance, still fills down happily and still produces a plausible-looking column of growing numbers.
A second test worth running. Change the rate in column C to $0$ . Every closing balance should collapse to a plain multiple of $\$2000$ , so period 4 reads $\$8000$ . If it does not, the interest column is feeding on something it should not be.
Once it is built, the spreadsheet answers questions the formula cannot show you at a glance: raise the contribution, lengthen the term, change the rate, and watch which of the three moves the final balance most.
To model an annuity in a spreadsheet, use one row per period with columns for period, opening balance, interest, contribution and closing balance, then fill down. The key formula is that each opening balance equals the previous closing balance. Check the model by matching the final closing balance to the FV formula.
Pause, copy the five cell formulas, and the two checks: the period-4 closing balance must match the formula, and setting the rate to zero must leave plain multiples of the contribution.
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Work annuity examples
Apply future value reasoning to worked savings scenarios.
Worked examples · reveal each step
PROBLEM 1 · FUTURE VALUE OF SUPERANNUATION
$300 deposited fortnightly into super at 5.4% p.a. compounded fortnightly for 20 years. Find the future value and total interest earned.
Interest earned = FV − total contributions. Compound interest nearly doubles the total!
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Compare present and future value
Use present value, avoid common traps and consolidate the lesson through independent practice.
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Present value, what is a future stream worth today?
core concept
FV = M × [(1 + r)ⁿ − 1] / r tells you what regular deposits grow to, but sometimes you need the reverse: what single lump sum today equals receiving $M$ per period for $n$ periods? That is present value, and the PV formula discounts each future payment back to today's dollars: PV = M × [1 − (1 + r)^(−n)] / r.
The present value of an ordinary annuity answers: "What single lump sum today is equivalent to receiving $M$ per period for $n$ periods at interest rate $r$?"
$$PV = M \times \frac{1 - (1+r)^{-n}}{r}$$
This is the reverse of the FV formula, we discount each future payment back to today.
Worked example: What lump sum today equals $500 monthly for 5 years at 6% p.a. compounded monthly?
$r = 0.06/12 = 0.005$. $n = 60$.
$PV = 500 \times \dfrac{1 - (1.005)^{-60}}{0.005} = 500 \times \dfrac{1 - 0.7414}{0.005} = 500 \times 51.73 = \$25{,}865$
A lump sum of $\$25,865$ today is equivalent to $\$500$/month for 5 years at this rate.
Time value of money: PV is always less than the sum of all payments ($500 × 60 = $30,000) because money received in the future is worth less than money today, future payments are "discounted" to account for the interest that could have been earned on a lump sum now.
Present value of an ordinary annuity: PV = M × [1 − (1 + r)^{−n}] / r. PV discounts all future payments back to today's dollars. Use when comparing a lump sum now versus regular payments over time.
Pause, copy PV = M × [1 − (1 + r)^(−n)] / r with all variable labels, and note: use PV when comparing a lump sum now against a stream of future payments into your book.
Fill the gap: For an annuity paying $M$ per period, if the interest rate per period is $r$ and there are $n$ periods, the present value formula is $PV = M \times \dfrac{1 - (1+r)^{-n}}{r}$. If $M = 100$, $r = 0.01$, and $n = 12$, the factor $\dfrac{1-(1.01)^{-12}}{0.01}$ equals approximately .
Common errors · the 3 traps that cost marks
Trap 01
Using annual rate instead of rate per period
If payments are monthly and the annual rate is 6%, you MUST use $r = 0.005$ (not 0.06) in the formula. Failing to divide by 12 gives a wildly incorrect answer, the most common annuity error in the HSC.
Trap 02
Confusing FV and PV situations
FV: you are building up savings (deposits going in, asking what they grow to). PV: you are drawing down / comparing to today's dollars (payments coming out, asking the equivalent lump sum today). Read the question carefully to identify which direction time is moving.
Trap 03
Confusing total payments with FV
Total payments = $M \times n$. This is always less than FV (the annuity formula includes compound interest on top). The difference (FV − M×n) is the total interest earned, a common sub-question in HSC problems.
Match each formula/concept to its description:
FV = M[(1+r)^n-1]/r
PV = M[1-(1+r)^-n]/r
r = annual rate / k
FV - M×n
Rate per period calculation
Total interest earned
Future value of regular deposits
Present value of future payments
Quick-fire practice · 2 activities
1
Calculate the future value and total interest for these annuities: (a) $\$150$ monthly at 3.6% p.a. compounded monthly for 5 years. (b) $\$800$ quarterly at 6% p.a. compounded quarterly for 10 years.
2
Is it better to receive $\$10,000$ today or $\$500$/month for 2 years at 6% p.a. compounded monthly? Use PV to decide. Show your working.
Top 3 list: List THREE real-world financial products or situations that use annuity mathematics. For each, state whether you would use the FV or PV formula, and briefly explain why.
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Revisit your thinking
It is NOT simply $\$2,400$. Each $\$200$ deposit earns compound interest for a different number of months, the first deposit earns interest for 12 months, the second for 11 months, and so on. Using the FV formula: $r = 0.004$, $n = 12$: $FV = 200 \times \dfrac{(1.004)^{12}-1}{0.004} = 200 \times 12.28 = \$2{,}456$. The extra $56 is interest. Over longer periods this effect becomes dramatic, this is the power of regular saving combined with compound interest.
What has changed in your understanding? What did you get right? What surprised you?
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Apply annuity models
Calculate future and present values, then justify decisions in exam-style questions.
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Multiple choice
+5 XP per correct · +25 XP all-correct
Pick your answer, then rate your confidence. That tells the system what to drill next.
Q1. $\$400$ deposited monthly at 6% p.a. compounded monthly. What value of $r$ should be used in the annuity formula?
Q2. The future value of an ordinary annuity is always:
Q3. $\$250$ deposited monthly at 4.8% p.a. compounded monthly for 2 years. What is $n$ in the FV formula?
Q4. An annuity due differs from an ordinary annuity because:
Q5. The present value of an ordinary annuity is always less than the total of all future payments because:
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Short answer
ApplyBand 42 marks
SA 1. (a) Find the future value of $250 monthly deposits at 4.8% p.a. compounded monthly for 4 years. (b) Find the total interest earned. (c) How much more would be earned if the rate were 6% p.a.? (2 marks)
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ApplyBand 42 marks
SA 2. Find the present value of $500 monthly for 5 years at 6% p.a. compounded monthly. Explain what this present value represents in practical terms. (2 marks)
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AnalyseBand 53 marks
SA 3. A person contributes $400/month to superannuation from age 25 to 65 (40 years) at 6% p.a. compounded monthly. (a) Find the future value. (b) If they delay starting until age 35 (30 years), what is the new FV? (c) Calculate the cost of the 10-year delay. (d) Why is the cost so large? (3 marks)
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Comprehensive answers (click to reveal)
MC 1, B: $r = 0.06/12 = 0.005$ per month. Always divide annual rate by periods per year.
MC 2, C: FV includes compound interest on top of all deposits, so it always exceeds the sum M×n.
MC 3, A: $n = 2 \times 12 = 24$ monthly payments.
MC 4, D: Annuity due: payments at the start of each period, giving one extra period of interest per payment.
MC 5, B: Time value of money: a dollar today is worth more than a dollar in the future because it can earn interest.
SA 2 (2 marks): $PV = 500 \times 51.73 = \$25{,}865$ [1]. This is the lump sum today that, invested at 6% compounded monthly, would provide exactly $500/month for 5 years [1].
SA 3 (3 marks): (a) $n=480$; $FV = 400 \times 1991.5 = \$796{,}600$ [0.5]. (b) $n=360$; $FV = 400 \times 1004.5 = \$401{,}800$ [0.5]. (c) Cost = $394{,}800$ [0.5]. (d) The 10 missing years lose not only $48{,}000$ in deposits but also $346{,}800$ in compound interest on those deposits and all the subsequent compounding, exponential growth makes early contributions enormously valuable [0.5].
Drill 2: $r=0.005$, $n=24$; $PV = 500 \times 22.56 = \$11{,}279$. Since $PV < \$10{,}000$... wait, $\$11{,}279 > \$10{,}000$, so the payment stream is better than the lump sum.
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Finish and choose your next step
Use the module challenge, mark the lesson complete and continue through Annuities.
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Take the full module quiz
quiz
A full module quiz covering every lesson in this module, not just this one. Set aside a decent block of time and treat it like a real assessment.
