IB Maths AI SLTopic 1 — Financial ApplicationsPaper 2GDC essential~7 min read
Annuities
An annuity is the opposite of a loan — you invest a lump sum (or pay in regularly), and the bank pays you back over time with added interest. Same TVM solver as amortisation, but the signs flip.
📘 What you need to know
Annuity = receive regular payments from a lump-sum investment, with interest. Common for retirement income.
Sign convention: PV is negative (you invest money out); PMT is positive (you receive money back).
PMT@ START for annuities — payment at the start of each period (slightly more favourable to you).
FV = 0 means the annuity exactly runs out after N payments.
Total received > original investment — the difference is the interest earned over the annuity’s life.
Sinking fund = a savings annuity. You DEPOSIT regularly (PMT negative) to grow a future lump sum (FV positive).
Amortisation vs Annuity — sign comparison
Variable
Amortisation (loan)
Annuity (income)
PV
positive — you receive the loan
negative — you invest the lump sum
PMT
negative — you pay back each period
positive — you receive each period
FV
0 (loan paid off)
0 (annuity used up)
PMT@
END
START
The flip rule: if you understood amortisation, just reverse the signs on PV and PMT and switch PMT@ to START. Everything else (N, I%, P/Y, C/Y) stays the same.
The TVM solver for annuities
PV in red because it’s entered as a NEGATIVE number (money you invest). PMT@ in amber because for annuities it’s START, not END.
Sinking funds — saving INTO an account
A sinking fund is an annuity in reverse: you deposit regularly to build up a target future amount. PMT is negative (money out), FV is positive (target). PMT@ is usually END (deposits at end of period).
🧭 Recipe — any annuity / sinking-fund problem
Identify the direction: are you RECEIVING money from a lump sum (annuity) or SAVING UP toward a goal (sinking fund)?
Set the signs: money you put IN = negative; money you get OUT or accumulate = positive.
Set PMT@: START for receiving-an-annuity; END for sinking-fund deposits.
Match P/Y and C/Y to the payment / compounding frequency.
Leave blank what’s unknown, press solve, then quote the answer as a positive amount.
Worked examples
WE 1
Find PMT — basic retirement annuity
Hassan invests $500 000 at retirement in an annuity paying 4% nominal annual, compounded monthly. He wants monthly income for 20 years. Find his monthly payment.
Marco invests €400 000 at 3.5% nominal annual, compounded monthly. He withdraws €2500 each month. How long, in years and months, will his annuity last?
Lin deposits $800 at the end of every month into a savings account paying 5% nominal annual, compounded monthly. Find the value of the account after 15 years.
TVM solver inputs (sinking fund — PMT negative, FV positive)N = 180 (15 × 12), I% = 5PV = 0 (no lump sum to start)PMT = −800, FV = ?P/Y = C/Y = 12, PMT@ = ENDGDC returnsFV = 213 831.16account value ≈ $213 831.16total deposits = 800 × 180 = $144 000. Extra $69 831 is the interest earned.
WE 5
Total received vs invested
For Hassan’s annuity in WE 1, find the total amount received over 20 years and the total interest earned.
Total receivedtotal = 240 × 3019.84 = $724 760.53Interest earnedinterest = total − initial investment = 724 760.53 − 500 000total received: $724 760.53, interest: $224 760.53Hassan ends up with ~45% more than he invested — the power of compound interest working in your favour.
WE 6
Different compounding vs payment frequency
Janelle invests $1 000 000 in an annuity paying 6% nominal annual interest, compounded annually, with monthly payouts for 25 years. Find her monthly payment.
TVM solver inputs (C/Y ≠ P/Y)N = 300 (25 × 12), I% = 6PV = −1 000 000, PMT = ?FV = 0, P/Y = 12, C/Y = 1, PMT@ = STARTGDC returnsPMT = 6315.47Janelle receives $6315.47/monthwhen compounding and payment frequencies differ, just set P/Y and C/Y separately on the GDC — it handles the conversion.
💡 Top tips
Annuity = opposite of loan: flip the signs on PV and PMT, switch PMT@ to START.
Write every TVM input on your paper — examiner method marks depend on it.
Quote answers as positive — the GDC’s sign is just bookkeeping.
⚠ Common mistakes
Same signs for PV and PMT: if both are negative (or both positive), the GDC will give nonsense. Always opposite signs.
PMT@ END for annuity: for receiving-an-annuity, it’s START. Wrong setting gives a small but examiner-detectable error.
Forgetting to convert months back to years and months: N comes out in periods (months) but the question often wants years and months.
Mixing up P/Y and C/Y: when they differ (like WE 6), each goes in a separate slot on the GDC.
That completes Financial Applications. Up next per the syllabus: Geometry & Trigonometry — starting with 3D coordinate geometry and distance / midpoint formulas.
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