Introducing Annuities — Future Value
Most people don't invest once and wait — they contribute regularly. Monthly into super, fortnightly into savings, annually into an education fund. An annuity is the mathematical model for this reality: regular payments plus compound interest, growing together into a future sum. You'll derive the formula from geometric sequences and see why it powers every retirement calculator on Earth.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
You deposit $500 at the end of every month into an account paying 0.5% per month. After 12 months, will your balance be: A) Exactly $6,000? B) Slightly more than $6,000? or C) Significantly more?
Make a prediction and explain your reasoning — no formula yet.
An annuity is a series of equal payments made at regular intervals, with each payment earning compound interest. The two most common types in the HSC:
Key facts
- The future value annuity formula
- The difference between ordinary annuity and annuity due
- That $r$ and $n$ must match the contribution frequency
Concepts
- How the annuity formula is derived from the GP sum
- Why regular contributions create accelerating growth
- The difference between total contributions and future value
Skills
- Calculate FV for any ordinary annuity
- Convert annual rates and years to match monthly/quarterly contributions
- Transpose the formula to find $a$, $r$, or $n$
- Model superannuation and savings scenarios
Consider $a$ deposited at the end of each period for $n$ periods at rate $r$ per period.
The last payment earns no interest. The second-last earns one period. The first earns $(n-1)$ periods of interest:
This is a geometric series with:
- First term: $a$
- Common ratio: $(1+r)$
- Number of terms: $n$
Using the GP sum formula $S_n = \dfrac{a(r^n - 1)}{r - 1}$ with ratio $(1+r)$:
Substitute into the sum formula and simplify — the denominator becomes simply $r$:
Each contribution earns a different number of periods of interest — the first earns the most, the last earns none.
FV annuity formula: $FV = a \times \dfrac{(1+r)^n - 1}{r}$; Derived from GP sum with first term $a$, ratio $(1+r)$, $n$ terms
Pause — copy the FV annuity formula $FV = a \times \dfrac{(1+r)^n - 1}{r}$ — derived from the GP sum with first term $a$ and ratio $(1+r)$ — into your book.
Did you get this? True or false: in the future value annuity formula, the annuity payments form a geometric series with common ratio $(1+r)$.
Worked examples · 3 in a row, reveal as you go
You deposit $300 at the end of each month into an account earning 4.8% p.a. compounded monthly. What is the balance after 5 years?
Jax contributes $450 fortnightly to his super fund returning 6.4% p.a. compounded fortnightly. He works for 25 years. Find (a) his balance at retirement and (b) how much is contributions vs interest.
Find the future value of $200 deposited at the end of each quarter for 8 years at 5.2% p.a. compounded quarterly.
Quick check: $500 is deposited monthly for 2 years at 6% p.a. compounded monthly. Which values are correct for the formula?
Common errors · the 3 traps that cost marks
Write the two-step process to convert an annual interest rate and number of years into the correct $r$ and $n$ for quarterly contributions. Give an example with 5% p.a. over 3 years.
Quick-fire practice · 4 drills
Find $FV$ for $a = \$400$, $r = 5\%$ per period, $n = 12$ periods.
Find $FV$ for $a = \$1{,}000$, $r = 6\%$ per period, $n = 20$ periods.
$\$250$ per month for 4 years at 3.6% p.a. compounded monthly. Find $FV$.
$\$600$/month for 35 years at 7% p.a. compounded monthly. What is $FV$ and how much is interest?
Fill in the blanks. The future value of an ordinary annuity with contribution $a$, rate per period $r$, and $n$ periods is: $FV = a \times \dfrac{(\underline{\phantom{1+r}})^n - \underline{\phantom{1}}}{r}$. The formula is derived from a __________ series with common ratio $(1+r)$.
Match each scenario to the correct $r$ and $n$ values.
Scenario A: Monthly contributions, 4% p.a., 3 years → $r = $ __ , $n = $ __
Scenario B: Quarterly contributions, 6% p.a., 5 years → $r = $ __ , $n = $ __
Scenario C: Fortnightly contributions, 5.2% p.a., 2 years → $r = $ __ , $n = $ __
Earlier you predicted whether $500/month at 0.5%/month for 12 months would give exactly $6,000, slightly more, or significantly more. The answer is B — slightly more than $6,000:
The interest is only $168 because the time is short and each contribution has limited time to compound. Over 10 years, interest would be $2,300. Over 30 years, $14,000. Time is the critical variable in annuity growth.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. $250 is deposited at the end of each month for 5 years into an account earning 6% p.a. compounded monthly. Find the future value and the total interest earned. (3 marks)
Q2. Ari wants a super balance of $600,000 at retirement in 30 years. The fund earns 6% p.a. compounded monthly. What monthly contribution $a$ does she need? (3 marks)
Q3. (a) $800 per month is invested for 20 years at 4.8% p.a. compounded monthly. Find the FV and interest earned. (b) Explain why the interest earned exceeds the total contributions. (4 marks)
Comprehensive answers (click to reveal)
Drill 1: $FV = 400 \times [(1.05)^{12}-1]/0.05 = \$6{,}398.51$ 2: $FV = 1{,}000 \times [(1.06)^{20}-1]/0.06 = \$36{,}785.59$ 3: $r=0.003$, $n=48$, $FV = \$12{,}726.86$ 4: $r=0.005833$, $n=420$, $FV \approx \$1{,}082{,}537$; contributions $= \$252{,}000$; interest $= \$830{,}537$
Match: A: $r=0.0\overline{3}$, $n=36$ | B: $r=0.015$, $n=20$ | C: $r=0.002$, $n=52$
Q1 (3 marks): $r = 0.06/12 = 0.005$, $n = 60$ [1]. $FV = 250 \times [(1.005)^{60}-1]/0.005 = \$17{,}442.51$ [1]. Interest $= \$17{,}442.51 - \$15{,}000 = \$2{,}442.51$ [1].
Q2 (3 marks): $600{,}000 = a \times [(1.005)^{360}-1]/0.005$ [1]. $a = 600{,}000/1{,}004.52 = \$597.30$/month [2].
Q3 (4 marks): (a) $r=0.004$, $n=240$. $FV = 800 \times [(1.004)^{240}-1]/0.004 = \$396{,}287.29$ [2]. Interest $= \$396{,}287.29 - \$192{,}000 = \$204{,}287.29$ [1]. (b) Later payments have up to 20 years to compound, generating interest on interest — the exponential effect of compounding over many periods [1].
Five timed annuity questions. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
Enter the arenaClimb platforms by answering annuity future value questions. Pool: lessons 1–7.
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