IB Maths AI HLFinancial ApplicationsPaper 2 & 3TVM solver~7 min read
Annuities
An annuity is the mirror image of a loan: instead of repaying money you borrowed, you invest a sum and receive a stream of regular payments back. Because interest is added along the way, the payments total more than the sum invested. The same GDC finance solver does the work — with the signs reversed.
📘 What you need to know
An annuity pays a sum back to you in regular instalments, with interest, instead of as a single lump sum.
You always receive more in total than the original sum — the extra is interest earned.
Solve with the GDC’s finance / TVM solver — the same fields as a loan.
Sign convention reversed from a loan: PV is negative (you invest it), PMT is positive (you receive it).
For annuities, payments are assumed at the start of each period unless told otherwise.
An annuity formula exists but is not examinable — use the GDC.
What is an annuity?
When a sum of money is owed to you — an inheritance, or the return on an investment — you can take it all at once as a lump sum, or have it paid back gradually in regular instalments with interest added. That stream of instalments is an annuity.
Because interest accumulates on the money still held, the instalments add up to more than the original sum. The longer the annuity runs, the larger that total becomes.
One lump sum goes in; a stream of regular payments comes back. Their total exceeds what was invested — the difference is the interest the annuity earns.
Using the GDC’s finance solver
Annuity questions use the same finance / TVM solver as loans. Enter every known value, leave the unknown blank, and let the GDC solve it. The fields are the same — N, I%, PV, PMT, FV, P/Y, C/Y — but the signs are flipped.
Signs are reversed from a loan: here PV is negative (you hand the money over) and PMT is positive (the money comes back to you). Set FV = 0 and assume PMT@ = START unless the question says otherwise.
Total received and the lump sum comparison
Once the GDC gives N and the payment, the total received is the number of payments times the payment. Comparing that total with the original lump sum shows how much extra the annuity delivers.
Total received from an annuity
total received = N × (payment)
interest earned = total received − amount investedthe total always exceeds the lump sum — the gap is the interest the annuity earns
Note: there is an annuity formula, FV = A × (1 + r)n − 1r, but it is not examinable — the GDC’s finance solver is all you need.
🧠Recipe — solving an annuity problem
Identify the annuity — the amount invested, the rate, the payment and the timeframe.
Open the finance / TVM solver and enter every known value.
Apply the signs: PV is negative (money you invest), PMT is positive (money you receive).
Set FV = 0, P/Y = C/Y = periods per year, and PMT@ = START unless told otherwise.
Leave the unknown blank and solve; write out every input for method marks.
Worked examples
WE 1
Finding the payment received
Maya invests $120 000 into an annuity paying a nominal annual rate of 4.8%, compounded monthly. She wants monthly payments for 15 years. Find the monthly payment she receives.
15 years monthly ⇒ N = 180; solve for PMTN = 180, I% = 4.8, PV = −120000, FV = 0, P/Y = C/Y = 12, PMT@ STARTGDC gives PMT = 932.77monthly payment = $932.77PV is negative — Maya hands the money over — so PMT comes back positive.
WE 2
How long the annuity lasts
An annuity of $80 000 pays a nominal annual rate of 3.6%, compounded monthly. The holder withdraws $700 at the start of each month. For how many years and months will the annuity last?
solve for NI% = 3.6, PV = −80000, PMT = 700, FV = 0, P/Y = C/Y = 12, PMT@ STARTN = 139.64 monthsconvert: 139.64 ÷ 12 = 11.64 years0.64 × 12 ≈ 8 months11 years and 8 monthslarger withdrawals would empty the annuity sooner.
WE 3
Finding the investment needed
How much must be invested now into an annuity paying a nominal annual rate of 5.4%, compounded monthly, to provide payments of $1500 at the start of each month for 12 years?
12 years monthly ⇒ N = 144; solve for PVN = 144, I% = 5.4, PMT = 1500, FV = 0, P/Y = C/Y = 12, PMT@ STARTGDC gives PV = −159430invest ≈ $159430PV comes out negative — report it as the positive amount to be invested.
WE 4
Total received and the excess
Sam invests $50 000 into an annuity at a nominal annual rate of 4.2%, compounded monthly, receiving payments at the start of each month for 8 years. (a) Find the monthly payment. (b) Find the total Sam receives. (c) How much more is this than the amount invested?
(a) 8 years monthly ⇒ N = 96; solve for PMTN = 96, I% = 4.2, PV = −50000, FV = 0, P/Y = C/Y = 12, PMT@ STARTPMT = 611.98(b) total = 96 × 611.98≈ $58750(c) excess = total − amount invested(a) $611.98 · (b) $58750 · (c) $8750the $8750 excess is the interest the annuity earned over the 8 years.
WE 5
Lump sum or annuity?
Priya is owed $90 000. She can take it as a lump sum now, or as an annuity at a nominal annual rate of 4%, compounded monthly, with payments at the start of each month for 10 years. (a) Find the monthly annuity payment. (b) Find the total she receives from the annuity. (c) How much more does the annuity give than the lump sum?
(a) 10 years monthly ⇒ N = 120; solve for PMTN = 120, I% = 4, PV = −90000, FV = 0, P/Y = C/Y = 12, PMT@ STARTPMT = 908.18(b) total = 120 × 908.18≈ $108982(c) extra = 108982 − 90000(a) $908.18 · (b) $108982 · (c) $18982the annuity gives $18982 more — the price of waiting is the reward of interest.
WE 6
Full question: a retirement annuity
On retirement, Diego invests €250 000 into an annuity returning a nominal annual rate of 5%, compounded monthly, with payments at the start of each month for 20 years. (a) Find the monthly payment. (b) Find the total amount Diego receives. (c) Find the total interest the annuity earns him.
(a) 20 years monthly ⇒ N = 240; solve for PMTN = 240, I% = 5, PV = −250000, FV = 0, P/Y = C/Y = 12, PMT@ STARTPMT = 1643.04(b) total = 240 × 1643.04≈ €394330(c) interest = 394330 − 250000(a) €1643.04 · (b) €394330 · (c) €144330over 20 years the annuity returns far more than the sum first invested.
💡 Top tips
For an annuity the signs flip from a loan: PV negative, PMT positive.
Money to you is positive, money from you is negative — the same rule, applied carefully.
Set PMT@ = START for annuities unless the question says end of period.
Count periods, not years: a 12-year monthly annuity has N = 144.
The annuity formula is not examinable — always use the GDC’s finance solver.
âš Common mistakes
Using loan signs — for an annuity PV is negative and PMT is positive, the reverse of a loan.
Leaving PMT@ on END when an annuity defaults to START.
Using years for N — multiply the number of years by the payments per year.
Reporting PV as negative — state the amount invested as a positive figure.
Confusing total received with interest — subtract the amount invested to get the interest.
That completes Financial Applications. Compound interest and depreciation use a formula; amortisation and annuities use the GDC’s finance solver — the same tool, with signs telling apart money borrowed, repaid, invested and received.
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