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.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
$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.
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.
Key facts
- FV formula for ordinary annuity
- Annuity due adjustment factor
- Formula for required payment $M$
- Total interest formula
Concepts
- Why early deposits matter most
- The compounding effect on regular payments
- Ordinary vs annuity due timing difference
Skills
- Calculate FV of any ordinary annuity
- Calculate FV of an annuity due
- Find the required regular payment
- Calculate total interest earned
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.
$r = 0.06 \div 12 = 0.005$, $n = 10 \times 12 = 120$
$FV = 200 \times \dfrac{(1.005)^{120} - 1}{0.005} = 200 \times \dfrac{1.8194 - 1}{0.005} = 200 \times 163.88 = \$32{,}776$
Total deposited $= 200 \times 120 = \$24{,}000$. Interest earned $= \$8{,}776$.
$$M = FV \times \frac{r}{(1+r)^n - 1}$$ 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.
What to write in your book
- $FV = M \times \dfrac{(1+r)^n - 1}{r}$ — ordinary annuity (payments at end of period).
- Convert annual rate: $r = r_{annual} \div \text{periods per year}$.
- Interest earned $= FV - (M \times n)$.
- Finding payment: $M = FV \times \dfrac{r}{(1+r)^n - 1}$.
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?
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.
$r = 0.004$, $n = 24$
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.
What to write in your book
- Annuity due: payments at the start of each period.
- $FV_{due} = FV_{ordinary} \times (1 + r)$.
- Each payment earns one extra period of interest — hence multiply by $(1+r)$.
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.
Worked examples · reveal each step
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.
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.
The most important variable in the FV formula is time. Early deposits compound for more periods, producing dramatically larger results.
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.
What to write in your book
- Starting early is the most powerful factor — 10 extra years can more than double FV.
- Each extra year compounds ALL previous payments for one more period.
- Always set $r = r_{annual} \div (\text{periods/year})$ and $n = \text{years} \times (\text{periods/year})$ before substituting.
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:
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?
Show answer
$FV = 150 \times \dfrac{(1.002077)^{650} - 1}{0.002077} \approx 150 \times 1273.6 = \$191{,}040$.
Total deposited $= 150 \times 650 = \$97{,}500$. Interest $= \$93{,}540$.
$FV_{due} = 191040 \times 1.002077 \approx \$191{,}437$.
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.
The final amount is significantly more than $2400. $FV = 100 \times \dfrac{(1.005)^{24} - 1}{0.005} = 100 \times 25.43 = \$2{,}543$. 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?
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:
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)
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)
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)
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) $\approx \$34{,}156$.
Five timed questions on future value, required payments, annuity due, and total interest. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms using future value calculations. Pool: lesson 5.
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