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Module 7 · L4 of 12 ~25 min MS12-5 ⚡ +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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Worksheets

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Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.

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Recall — your gut answer first

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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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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.

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

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.

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: $FV_{\text{due}} = FV_{\text{ordinary}} \times (1+r)$.

What to write in your book

Annuity: equal payments (M) at equal time intervals.
Ordinary: end of period (mortgages, loans). Annuity due: start of period (rent, leases).
Variables: M = payment, r = rate per period, n = number of payments.

Quick check: Monthly mortgage repayments are an example of which type of annuity?

Future value — what will regular savings become?

core concept

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.

What to write in your book

FV of ordinary annuity: $FV = M \times \dfrac{(1+r)^n - 1}{r}$
$r$ = rate per period (annual rate ÷ periods per year). $n$ = total number of payments.
Interest earned = FV − (M × n).

True or false: In an ordinary annuity, the last payment deposited earns the most compound interest over the investment period.

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.

$r = 0.054/26 = 0.002077$ per fortnight, $n = 20 \times 26 = 520$

There are 26 fortnights per year; 20 years gives 520 payments

Present value — what is a future stream worth today?

core concept

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.

What to write in your book

PV of ordinary annuity: $PV = M \times \dfrac{1 - (1+r)^{-n}}{r}$
PV is always less than the total of all payments (due to time value of money).
Use PV to compare "lump sum today" vs "regular payments over time".

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.

What to write in your book

Always use rate per period: r = annual rate ÷ periods per year.
FV = future growth of deposits. PV = current value of future payments.
Interest earned = FV − (M × n).

Match each formula/concept to its description:

Quick-fire practice · 2 activities

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.

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.

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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Multiple choice

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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:

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 1 (2 marks): (a) $r=0.004$, $n=48$. $FV = 250 \times 52.88 = \$13{,}220$ [0.5]. (b) Interest = $1{,}220$ [0.5]. (c) At 6%: $r=0.005$; $FV = 250 \times 54.10 = \$13{,}525$; Extra = $305$ [1].

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 1: (a) $FV = \$9{,}945$; interest = $945$. (b) $FV = \$43{,}678$; interest = $11{,}678$.

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.

Boss battle · The Annuity Calculator

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Five timed questions on annuity definitions, FV and PV calculations, and time value of money. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.

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Science Jump · platform challenge

Climb platforms by answering questions on annuities. Pool: lesson 4.

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Module overview · Maths Standard