Should you pay rent at the start or end of the month? The answer is yes, it matters, and the difference can be worth thousands of dollars over a lifetime. One simple adjustment, $(1+r)$, converts ordinary annuity calculations into annuity due calculations. Timing is everything in financial mathematics.
Today's hook, A gym charges $\$500$/month. Ordinary: you pay at the end of each month. Annuity due: you pay at the start. At 6% p.a., the gym earns $\$56$ more in present value just from the timing shift. They've priced it in. Have you noticed?
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Get oriented
Recall what you already know, meet the key ideas and settle the terms.
Worksheets
Practise this lesson
Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
A car lease offers two plans: Plan A$400/month paid at the end of each month for 36 months. Plan B$390/month paid at the beginning of each month for 36 months. At 7.2% p.a. compounded monthly, predict which plan has the lower present value before calculating.
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02
The one adjustment that changes everything
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An ordinary annuity has payments at the end of each period, the default in most HSC questions. An annuity due has payments at the beginning of each period. Rent, leases, insurance premiums, and gym memberships are typically structured this way.
The adjustment is elegantly simple, it works for both PV and FV:
Every payment in an annuity due sits one period earlier than the corresponding payment in an ordinary annuity. Earlier payment means one extra period of interest earned (or discounted). So multiply the ordinary result by $(1+r)$.
$\text{Due} = \text{Ordinary} \times (1+r)$
Ordinary annuity
Payments at end of period. Default unless HSC says otherwise. $n$ payments, first payment at end of period 1.
Annuity due
Payments at beginning of period. Real-world: rent, insurance, leases. Same $n$ payments, just shifted one period earlier.
The single rule
Due $=$ Ordinary $\times (1+r)$. Works for both PV and FV. One rule, two formulas covered.
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What you'll master
Know
Key facts
The definition and formula for annuity due
The adjustment factor $(1+r)$ for timing
When HSC questions specify ordinary vs due
Understand
Concepts
Why beginning-of-period payments accumulate more
Why rent, leases and insurance are often annuity due
The symmetry between PV and FV adjustments
Can do
Skills
Identify whether a scenario is ordinary or due
Calculate FV and PV for annuity due
Compare payment plans with different timing
Transpose formulas to find $a$, $n$ or $r$
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Key terms
Ordinary annuityAn annuity where payments occur at the end of each period.
Annuity dueAn annuity where payments occur at the beginning of each period.
Payment timingWhether each payment falls at the start or end of a compounding period.
$(1+r)$ adjustmentThe multiplier that converts ordinary annuity PV or FV into the due equivalent.
Advance paymentA payment made at the beginning of a period, characteristic of annuity due.
Arrears paymentA payment made at the end of a period, characteristic of ordinary annuity.
Read the payment timeline
See what moves when payments fall at the start of a period instead of the end.
Beyond the syllabus. The syllabus specifies end-of-period payments (ordinary annuities) and does not explicitly require annuities due. Payment timing is useful understanding — treat the annuities due work as extension, and expect the exam to model ordinary annuities.
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Ordinary vs Annuity Due, the timeline
core concept
The number of payments $n$ does not change between ordinary and due. What changes is when each payment falls. In an annuity due, every payment earns exactly one extra period of interest compared to the ordinary case.
Each due payment (amber) sits exactly one period to the left of the corresponding ordinary payment (grey). That one period earns extra interest.
$$FV_{\text{due}} = a \times \frac{(1+r)^n - 1}{r} \times (1+r)$$
$$PV_{\text{due}} = a \times \frac{1 - (1+r)^{-n}}{r} \times (1+r)$$
Real-world example. A $500/month gym membership paid at the start of each month for 2 years at 6% p.a. compounded monthly has $PV_{\text{ord}} = 500 \times \frac{1-(1.005)^{-24}}{0.005} = \$11{,}290.11$ then $PV_{\text{due}} = 11{,}290.11 \times 1.005 = \$11{,}346.56$. The gym collects $56 more in present value terms from the timing shift alone.
Ordinary annuity: payments at end of period; default in HSC unless stated otherwise; Annuity due: payments at beginning of period; real examples: rent, insurance, leases
Pause, copy the distinction: ordinary annuity = payments at end of period (HSC default); annuity due = payments at beginning of period (multiply the ordinary FV/PV by $(1+r)$), into your book.
Did you get this? True or false: an annuity due has more payments than an ordinary annuity with the same $n$.
Work the examples
Follow three worked problems across ordinary annuities and annuities due.
Spotting annuity due in HSC questions
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HSC clue words, how to identify annuity due
exam technique
We just saw that ordinary annuities pay at the end of each period, while annuity due pays at the beginning, and this shifts every future value by a factor of $(1+r)$. That raises a question: how do you tell from HSC question wording which type you are dealing with, when the words "annuity due" are rarely used? This card answers it → look for "beginning of each period", "in advance", "upfront", or "first payment immediately"; default to ordinary annuity if timing is unspecified.
HSC questions rarely say "annuity due" explicitly. Train yourself to spot these trigger phrases:
Clue 01
"Payments at the beginning of each…"
The classic annuity due signal. If you see "beginning", apply the $(1+r)$ multiplier immediately.
Clue 02
"Rent paid in advance" / "Insurance upfront"
"In advance" or "upfront" means the first payment is immediate, start of period. Classic annuity due structure.
If the first payment happens right now (time $= 0$), that's an annuity due. Compare: ordinary annuity first payment is at time $= 1$.
Default rule. If the question does not specify timing, assume ordinary annuity (end of period). This is the HSC convention. Only apply the $(1+r)$ multiplier when timing is explicitly specified as beginning-of-period.
HSC annuity due signals: "beginning of each period", "in advance", "upfront", "first payment immediately"; Default assumption when timing is not stated: ordinary annuity (end of period)
Pause, copy the HSC annuity-due trigger phrases ("beginning of each period", "in advance", "upfront", "first payment immediately") and the default rule (no timing stated = ordinary annuity) into your book.
Quick check: Which scenario is an annuity due?
Worked examples · 3 in a row, reveal as you go
PROBLEM 1 · FV OF ANNUITY DUE
An investment scheme accepts deposits of $500 at the beginning of each month for 15 months. The account earns 6% p.a. compounded monthly. Find the future value.
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$r = 0.06/12 = 0.005$, $n = 15$, $a = 500$
Convert annual rate to monthly and identify parameters.
Multiply by $(1+r) = 1.005$ to convert to annuity due. This is 0.5% higher, exactly $r$.
PROBLEM 2 · PV OF ANNUITY DUE (insurance)
An insurance policy requires premiums of $250 at the beginning of each quarter for 5 years. The discount rate is 8% p.a. compounded quarterly. Find the present value of the premium stream.
Multiply by $(1+r) = 1.02$. The insurer collects $81.76 more in PV from getting payments at the start of each quarter.
PROBLEM 3 · COMPARING PAYMENT PLANS
A landlord offers: pay $\$1500$ at the end of each month, or $\$1470$ at the beginning. At 4.8% p.a. compounded monthly, which is better for the tenant?
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$r = 0.048/12 = 0.004$. Find PV of each plan over 12 months.
Convert rate to monthly. PV is the correct measure to compare since amounts differ.
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Plan A (ordinary): $PV = 1500 \times \dfrac{1-(1.004)^{-12}}{0.004} = \$17{,}613.52$
Standard ordinary annuity PV. $n = 12$ monthly periods.
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Plan B (due): $PV = 1470 \times \dfrac{1-(1.004)^{-12}}{0.004} \times 1.004 = \$17{,}306.26$
Plan B PV $= \$17{,}306.26 < \$17{,}613.52$. Plan B saves the tenant $307 in today's money despite being paid at the start of each month.
Try this now: A student pays $\$150$ at the beginning of each month for a tutoring package over 12 months. The rate is 4.8% p.a. compounded monthly. Find $PV_{\text{due}}$ to the nearest cent.
Fix the one-period slips that timing errors cause.
Common errors · the 3 traps that cost marks
Trap 01
Thinking annuity due has an extra payment
The number of payments $n$ is the same. Annuity due just shifts each payment one period earlier. Adding an extra payment is a completely different (and wrong) calculation.
Trap 02
Memorising separate formulas
You only need the ordinary annuity formula plus the $(1+r)$ multiplier. Students who try to memorise four separate formulas often mix them up under exam pressure.
Trap 03
Applying the multiplier to $a$ not to the result
The $(1+r)$ multiplies the entire ordinary annuity value, not just the payment amount $a$. Get the ordinary value first, then multiply by $(1+r)$.
Fill in the blank: To convert an ordinary annuity future value to an annuity due future value, you multiply the ordinary result by .
Drill it and revisit
Run the drill, then return to your opening answer and name what has changed.
Quick-fire practice · 4 timing problems
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Classify: "Insurance premiums of $80 paid upfront each month.", Ordinary or due?
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$FV_{\text{ord}} = \$8{,}000$, $r = 0.01$ per period. Find $FV_{\text{due}}$.
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$a = \$1{,}000$, $r = 3\%$ per period, $n = 30$. By what percentage is $FV_{\text{due}}$ higher than $FV_{\text{ord}}$?
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Create a scenario where the timing difference between ordinary and due is worth more than $\$500$ in PV terms. State all parameters.
Odd one out: Three of these are examples of annuity due. Which one is an ordinary annuity?
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Revisit your thinking
Earlier you predicted which car lease plan had the lower present value. The answer: Plan A (ordinary, $\$400$/month end): $PV = 400 \times \frac{1-(1.006)^{-36}}{0.006} = \$12{,}822.68$. Plan B (due, $\$390$/month start): $PV = 390 \times \frac{1-(1.006)^{-36}}{0.006} \times 1.006 = \$12{,}444.26$. Plan B is cheaper by $378 the $10/month saving and the timing both reduce PV.
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Practise payment timing
Write full short-answer responses, then check them against the model answers.
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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. Each retry pulls a fresh mix from the bank.
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Short answer
ApplyBand 43 marks
Q1. A savings plan accepts deposits of $\$500$ at the beginning of each quarter for $2\tfrac{1}{2}$ years. The interest rate is 12% p.a. compounded quarterly. (a) State $r$ and $n$ for this annuity. (b) Find $FV_{\text{ord}}$. (c) Find $FV_{\text{due}}$. (3 marks)
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ApplyBand 43 marks
Q2. A landlord requires rent of $800/month paid at the beginning of each month. The discount rate is 6% p.a. compounded monthly. (a) Find the ordinary annuity PV over 48 months. (b) Hence find the annuity due PV. (c) Explain why the due PV is higher than the ordinary PV for the same payment amount. (3 marks)
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AnalyseBand 54 marks
Q3. A car lease offers two plans at 7.2% p.a. compounded monthly over 36 months: Plan A, $\$400$/month at end. Plan B, $\$390$/month at start. (a) Calculate $PV_A$. (b) Calculate $PV_B$. (c) Which plan is better for the lessee? Justify with reference to both timing and payment amount. (d) Explain why the $\$10$ monthly saving combined with beginning-of-month timing results in a saving of more than $\$360$ in PV terms. (4 marks)
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📖 Comprehensive answers (click to reveal)
Drill 1: Annuity due (paid upfront/in advance). 2: $8{,}000 \times 1.01 = \$8{,}080$. 3: Due is exactly 3% higher, always equal to $r$. 4: Sample: $a = \$2{,}000$/month, $n = 60$, $r = 1\%$. Ordinary PV = $\$89{,}910$; Due PV = $\$90{,}809$. Difference = $899 > \$500$.
Q2 (3 marks): (a) $r = 0.005$, $n = 48$. $PV_{\text{ord}} = 800 \times \frac{1-(1.005)^{-48}}{0.005} = \$34{,}042.55$ [1]. (b) $PV_{\text{due}} = 34{,}042.55 \times 1.005 = \$34{,}212.76$ [1]. (c) Beginning-of-month payments are discounted one period less (each payment sits one period closer to today), so each has a higher present value, raising the total PV [1].
Q3 (4 marks): (a) $r = 0.006$. $PV_A = 400 \times \frac{1-(1.006)^{-36}}{0.006} = \$12{,}822.68$ [1]. (b) $PV_B = 390 \times \frac{1-(1.006)^{-36}}{0.006} \times 1.006 = \$12{,}444.26$ [1]. (c) Plan B, lower PV means the lessee pays less in today's money [1]. (d) The $\$10$ monthly saving reduces PV by $\approx \$320$; the beginning-of-month timing further increases the effective cost by $(1.006) - 1 \approx 0.6\%$ of Plan B's ordinary PV $\approx \$75$. The net effect is a $\$378$ saving, which exceeds $\$360 = 36 \times \$10$ because the timing also compounds across all 36 periods [1].
Finish the lesson
Take the module quiz, mark the lesson complete and move on.
01
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.
