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Module 7 · L5 of 12 ~25 min MST-12-S2-03 ⚡ +50 XP available

Future Value of Annuities

Every dollar you save today is a seed that grows into a forest tomorrow. The future value of an annuity tells you exactly how large that forest will be. Whether you are calculating your superannuation balance at retirement, the value of regular investments, or how much a savings plan will accumulate, the future value formula gives you the answer. It is the mathematical engine behind every retirement calculator, every investment projection, and every financial plan. Master it, and you can predict your financial future with precision.

Today's hook, $\$100$ is deposited at the end of each month for 2 years at 6% p.a. compounded monthly. Without calculating, will the final amount be close to $\$2400$, significantly more, or significantly less? Why?

0/5QUESTS

Orient to future value

Connect regular saving to the lesson goals, formula and essential language.

Worksheets

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Think First, your gut answer first

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$\$100$ is deposited at the end of each month for 2 years at 6% p.a. compounded monthly. Without calculating, do you think the final amount will be close to $\$2400$, significantly more, or significantly less?

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 is a series of equal regular payments. When saving or investing, you want the future value what your payments are worth at the end.

Ordinary annuity: payments are made at the end of each period. This is the standard assumption unless stated otherwise.

Annuity due: payments are made at the start of each period, earning one extra period of interest. Multiply the ordinary FV by $(1+r)$ to convert.

The FV formula for an ordinary annuity and its variables.

Interest earned $= FV - (M \times n)$. To find the required payment: $M = FV \times \dfrac{r}{(1+r)^n - 1}$

Convert rate first

Always divide the annual rate by the number of periods per year. Monthly: $r = r_{annual}/12$. Fortnightly: $r = r_{annual}/26$.

Count periods carefully

Monthly for 10 years: $n = 120$. Quarterly for 8 years: $n = 32$. Weekly for 5 years: $n = 260$.

Time is everything

Starting just 5 years earlier can more than double your final balance. Early payments compound for longer, time is the most powerful variable.

What you will master

Know

Key facts

FV formula for ordinary annuity
Annuity due adjustment factor
Formula for required payment $M$
Total interest formula

Understand

Concepts

Why early deposits matter most
The compounding effect on regular payments
Ordinary vs annuity due timing difference

Can do

Skills

Calculate FV of any ordinary annuity
Calculate FV of an annuity due
Find the required regular payment
Calculate total interest earned

Key terms

AnnuityA series of equal payments made at regular intervals over a fixed period.

Future Value (FV)The total accumulated value of all payments and interest at the end of the investment period.

Ordinary AnnuityPayments made at the end of each period, the standard assumption.

Annuity DuePayments made at the start of each period, earning one extra period of interest.

Period interest rate (r)Annual rate divided by the number of compounding periods per year.

Number of periods (n)Total count of payment intervals, years × payments per year.

Calculate future value

Convert rates and periods correctly, then apply the ordinary annuity formula.

Beyond the syllabus. Future value, total interest, required contributions and superannuation decisions are core, worked through future-value interest factors. Formula rearrangement, annuity due calculations, and logarithmic or geometric series methods for finding n are extension — the exam supplies factor tables and expects table or spreadsheet reasoning.

The future value formula, building wealth one payment at a time

core concept

The future value of an ordinary annuity is:

$$FV = M \times \frac{(1+r)^n - 1}{r}$$

Where $M$ is the regular payment, $r$ is the interest rate per period, and $n$ is the number of periods.

Why does this formula work? Each payment earns a different amount of interest, the first payment earns interest for $n-1$ periods, the second for $n-2$, and so on. Adding all these up gives a geometric series that simplifies to the formula above.

Worked example: $200 per month at 6% p.a. compounded monthly for 10 years.
$r = 0.06 \div 12 = 0.005$, $n = 10 \times 12 = 120$
$FV = 200 \times \dfrac{(1.005)^{120} - 1}{0.005} = 200 \times \dfrac{1.8194 - 1}{0.005} = 200 \times 163.88 = \$32{,}776$
Total deposited $= 200 \times 120 = \$24{,}000$. Interest earned $= \$8{,}776$.

Finding the required payment: Rearrange the formula to solve for $M$:
$$M = FV \times \frac{r}{(1+r)^n - 1}$$ Example: Target $50{,}000 in 8 years at 5.4% p.a. compounded monthly.
$r = 0.0045$, $n = 96$. $M = 50000 \times \dfrac{0.0045}{(1.0045)^{96}-1} = \dfrac{225}{0.5386} = \$417.75$/month.

FV = M × [(1 + r)^n − 1] / r. Convert the annual rate to a per-period rate (r = annual ÷ frequency) and count total periods (n = years × frequency). Each deposit earns compound interest for a different number of periods.

Pause, copy FV = M × [(1 + r)ⁿ − 1] / r, the rate conversion rule (r = annual rate ÷ periods per year), and the period count rule (n = years × periods per year) into your book.

Quick check: $\$300$ is invested monthly at 6% p.a. compounded monthly for 5 years. What is the value of $r$ used in the FV formula?

Compare payment timing

Explain why an annuity due earns one additional period of interest.

Annuity due, paying at the start of each period

core concept

The ordinary FV formula FV = M × [(1 + r)ⁿ − 1] / r assumes each payment is made at the end of the period, so the final payment earns no interest. An annuity due pays at the start of each period, meaning every payment earns one extra period of compound interest, the adjustment is simply FV_due = FV_ordinary × (1 + r).

When payments are made at the beginning of each period, each payment earns one extra period of interest compared to an ordinary annuity. The relationship is:

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

The factor $(1+r)$ accounts for the fact that every payment earns one extra compounding period.

Worked example: $300 per month at 4.8% p.a. compounded monthly for 2 years, payments at the start of each month.
$r = 0.004$, $n = 24$
Ordinary $FV = 300 \times \dfrac{(1.004)^{24} - 1}{0.004} = 300 \times 25.03 = \$7{,}509$
$FV_{due} = 7509 \times 1.004 = \$7{,}539$
The extra $30 comes from each of the 24 payments earning one additional month of interest.

Key insight: The difference between ordinary and due is usually small (about one period's interest on the total). In the HSC, read the question carefully, "end of period" means ordinary; "start of period" or "beginning of period" means due.

Annuity due pays at the start of each period, so each payment earns one extra period of interest: FV_due = FV_ordinary × (1 + r). Annuity due always produces a larger future value than ordinary for the same M, r, n.

Pause, copy FV_due = FV_ordinary × (1 + r) and explain why annuity due always gives a larger result: every payment earns one extra period of compound interest into your book.

True or false: An annuity due always has a higher future value than an ordinary annuity with the same payment, rate and number of periods.

Work future value examples

Apply the formula to savings plans and required-payment problems.

Worked examples · reveal each step

PROBLEM 1 · CALCULATING FUTURE VALUE

Sarah deposits $180 per month into a savings account paying 5.4% p.a. compounded monthly for 6 years. Find the future value and total interest earned.

$r = 0.054 \div 12 = 0.0045$, $\quad n = 6 \times 12 = 72$

Convert annual rate to monthly rate; count total periods

PROBLEM 2 · FINDING REQUIRED PAYMENT

Sarah wants $100,000 in her superannuation in 15 years. Her fund earns 7.2% p.a. compounded monthly. Find her required monthly contribution and total interest earned.

$r = 0.072 \div 12 = 0.006$, $\quad n = 15 \times 12 = 180$

Convert rate and count periods

Analyse the power of starting early

Compare contribution timelines and quantify the effect of additional compounding periods.

The power of starting early

core concept

Annuity due vs ordinary is a modest adjustment of ×(1 + r). The far bigger lever is n, the total number of periods. Because (1 + r)ⁿ grows exponentially with n, starting contributions even ten years earlier can nearly double the final balance for the same monthly payment amount.

The most important variable in the FV formula is time. Early deposits compound for more periods, producing dramatically larger results.

Compare: Both investors contribute $200/month at 6% p.a. compounded monthly.
Investor A starts at age 25 and contributes for 40 years (to age 65).
Investor B starts at age 35 and contributes for 30 years (to age 65).

Investor A: $FV = 200 \times \dfrac{(1.005)^{480}-1}{0.005} \approx \$400{,}290$
Investor B: $FV = 200 \times \dfrac{(1.005)^{360}-1}{0.005} \approx \$200{,}900$

Starting 10 years earlier (and contributing only $24{,}000 more) more than doubles the final balance.

This explains why financial advisers emphasise starting early. Delaying contributions by even 5 years costs far more than reducing contributions by 5%, time cannot be recovered.

Trap 01

Forgetting to convert the rate

Always divide the annual rate by the compounding frequency. Using 6% instead of 0.5% per month gives a wildly wrong answer.

Trap 02

Miscounting periods

Monthly for 5 years is 60 periods, not 5. Set up $n = \text{years} \times \text{periods per year}$ before substituting.

Trap 03

Ordinary vs annuity due

If the question says "start of period", calculate $FV_{ordinary}$ first, then multiply by $(1+r)$. Do not skip this step.

Starting contributions earlier dramatically increases FV because of compound interest over more periods. An extra 10 years of contributions can nearly double the final balance, even if the monthly payment is the same.

Pause, copy the starting-early principle (extra years compound exponentially, not linearly) and write one numerical comparison showing what happens to FV when n increases by 10 into your book.

Fill the gap: $\$500$ is deposited quarterly at 8% p.a. compounded quarterly for 3 years. The number of periods $n$ is .

Quick-fire practice · 2 activities

Calculate the future value of each annuity: (a) $\$100$/month at 4.8% compounded monthly for 5 years. (b) $\$500$/quarter at 6% compounded quarterly for 8 years. (c) $50/week at 5.2% compounded weekly for 10 years.

A 20-year-old and a 30-year-old both want $1 million at age 65, earning 7% p.a. compounded monthly. Find each person's required monthly contribution. How much more does the 30-year-old pay in total?

Match each term to its description:

Total number of payment periods
Total value at the end of the investment
Regular payment amount
Interest rate per compounding period

Consolidate future value

Solve an extended calculation and revisit the opening prediction.

Extended calculation, put it all together

apply

Starting earlier increases FV dramatically because n is an exponent in (1 + r)ⁿ. When a question gives you the target FV and asks for the required deposit M or the number of periods n, rearrange FV = M × [(1 + r)ⁿ − 1] / r to isolate the unknown, remembering to convert the annual rate to the correct per-period rate first.

Tom deposits $150 fortnightly at 5.4% p.a. compounded fortnightly for 25 years.

Find the future value.
Find total interest earned.
If instead he pays at the start of each fortnight, what is the new FV?

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Show answer

$r = 0.054 \div 26 = 0.002077$, $n = 650$.
$FV = 150 \times \dfrac{(1.002077)^{650} - 1}{0.002077} \approx 150 \times 1373.2 = \$205{,}980$.
Total deposited $= 150 \times 650 = \$97{,}500$. Interest $= \$108{,}480$.
$FV_{due} = 205980 \times 1.002077 \approx \$206{,}408$.

Multi-step FV problems may require finding n (e.g., doubling time) or M (required deposit). Rearrange FV = M × [(1+r)^n−1]/r for the unknown. Always convert the annual rate to per-period before substituting.

Pause, copy how to rearrange FV = M × [(1 + r)ⁿ − 1] / r to solve for M, and note that solving for n requires logarithms or trial-and-error on a table into your book.

Top 3 list: List THREE real-world situations where you would use the future value of an annuity formula. For each, state what $M$, $r$, and $n$ would represent.

Revisit your thinking

The final amount is significantly more than $\$2{,}400$. $FV = 100 \times \dfrac{(1.005)^{24} - 1}{0.005} = 100 \times 25.43 = \$2{,}543$. The extra $\$143$ is interest earned. Even with small payments and a modest rate, compound interest on regular deposits adds up. This is why consistent saving, even in small amounts, outperforms sporadic large deposits over time.

What has changed in your understanding? What did you get right? What surprised you?

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Apply future value

Calculate annuity values and explain how rate, time and payment timing affect the result.

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. $\$200$ is deposited monthly at 6% p.a. compounded monthly for 10 years. What is $n$?

Q2. An ordinary annuity has $FV = \$10{,}000$. The same annuity paid at the start of each period (annuity due) has a rate of $r = 0.004$ per period. What is $FV_{due}$?

Q3. $100/month at 4.8% p.a. compounded monthly for 5 years. Which calculation gives FV?

Q4. Emma wants $\$30{,}000$ in 6 years. Her account pays 5.4% p.a. compounded monthly. Which formula finds her required monthly payment $M$?

Q5. Total interest earned on an annuity where $M = \$250$, $n = 48$, and $FV = \$13{,}200$ is:

Short answer

ApplyBand 42 marks

SA 1. (a) Find the future value of $180 monthly deposits at 5.4% p.a. compounded monthly for 6 years. (b) Find the total interest. (c) If deposits were made at the start of each month instead, what would the FV be? (2 marks)

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ApplyBand 42 marks

SA 2. Emma needs $\$60{,}000$ in 10 years for a house deposit. Her savings account pays 4.8% p.a. compounded monthly. (a) Find her required monthly contribution. (b) If she can only afford $\$350$/month, how much will she have after 10 years? (2 marks)

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AnalyseBand 53 marks

SA 3. Person A contributes $\$500$/month from age 25 to 35 (10 years), then stops. Person B contributes $\$500$/month from age 35 to 65 (30 years). Both earn 7% p.a. compounded monthly. (a) Find Person A's balance at age 35. (b) Find what Person A's balance grows to by age 65 (no more contributions, just compounding). (c) Find Person B's balance at age 65. (d) Who wins, and why? (3 marks)

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Comprehensive answers (click to reveal)

MC 1, C: $n = 10 \times 12 = 120$ periods.

MC 2, B: $FV_{due} = 10000 \times 1.004 = \$10{,}040$.

MC 3, A: $r = 4.8\% \div 12 = 0.004$, $n = 60$. Formula A is correct.

MC 4, D: Payment formula is $M = FV \times r / [(1+r)^n - 1]$.

MC 5, B: Interest $= FV - (M \times n) = 13200 - 250 \times 48 = 13200 - 12000 = \$1{,}200$.

SA 1 (2 marks): (a) $r=0.0045$, $n=72$. $FV = 180 \times 83.28 \approx \$14{,}990$ [0.5]. (b) Deposited $= \$12{,}960$. Interest $= \$2{,}030$ [0.5]. (c) $FV_{due} = 14990 \times 1.0045 \approx \$15{,}057$ [1].

SA 2 (2 marks): (a) $M = 60000 \times 0.004 / [(1.004)^{120}-1] = 240/0.601 \approx \$399.33$/month [1]. (b) $FV = 350 \times 150.1 \approx \$52{,}535$ [1].

SA 3 (3 marks): (a) $FV_{A,35} \approx \$86{,}100$ [1]. (b) $A_{65} = 86100 \times (1.005833)^{360} \approx \$699{,}000$ [1]. (c) $FV_B \approx \$609{,}500$ [0.5]. (d) Person A wins despite contributing for 20 fewer years, the 30 extra years of compounding on their early balance is decisive [0.5].

Drill 1: (a) $\approx \$6{,}773$. (b) $\approx \$20{,}661$. (c) $\approx \$34{,}156$.

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