Annuities Due & Payment Timing
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.
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.
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 $(1+r)$.
Key facts
- The definition and formula for annuity due
- The adjustment factor $(1+r)$ for timing
- When HSC questions specify ordinary vs due
Concepts
- Why beginning-of-period payments accumulate more
- Why rent, leases and insurance are often annuity due
- The symmetry between PV and FV adjustments
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 $r$
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.
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 $n$.
Spotting annuity due in HSC questions
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:
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
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.
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.
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?
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.
Show answer
$r = 0.004$, $n = 12$. $PV_{\text{ord}} = 150 \times \frac{1-(1.004)^{-12}}{0.004} = \$1{,}761.35$. $PV_{\text{due}} = 1{,}761.35 \times 1.004 = \mathbf{\$1{,}768.40}$.
Common errors · the 3 traps that cost marks
Fill in the blank: To convert an ordinary annuity future value to an annuity due future value, you multiply the ordinary result by .
Quick-fire practice · 4 timing problems
Classify: "Insurance premiums of $80 paid upfront each month." — Ordinary or due?
$FV_{\text{ord}} = \$8{,}000$, $r = 0.01$ per period. Find $FV_{\text{due}}$.
$a = \$1{,}000$, $r = 3\%$ per period, $n = 30$. By what percentage is $FV_{\text{due}}$ higher than $FV_{\text{ord}}$?
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?
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.
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. A savings plan accepts deposits of $500 at the beginning of each quarter for $2\frac{1}{2}$ 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)
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)
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)
📖 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 = $899 > \$500$.
Q1 (3 marks): (a) $r = 0.12/4 = 0.03$, $n = 10$ [1]. (b) $FV_{\text{ord}} = 500 \times \frac{(1.03)^{10}-1}{0.03} = \$5{,}731.94$ [1]. (c) $FV_{\text{due}} = 5{,}731.94 \times 1.03 = \$5{,}903.90$ [1].
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].
Five timed questions on annuity due timing, PV and FV adjustments, and payment comparison. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms using annuity due timing, PV adjustments, and payment comparisons. Pool: lessons 1–9.
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