IB Maths AI SL Topic 1 — Financial Applications Paper 2 GDC essential ~7 min read

Amortisation

Amortisation means paying back a loan in fixed regular payments. The GDC’s TVM (time value of money) solver does all the heavy lifting — you just enter what you know and leave blank what you want.

📘 What you need to know

Amortisation = paying off a loan over time with fixed regular payments (e.g. a mortgage).
Each payment covers (i) interest on the outstanding balance and (ii) some of the principal. Balance decreases each period.
Total paid > amount borrowed — the difference is the interest paid to the lender.
Use your GDC’s TVM/finance solver — fill all known variables, leave blank the unknown, the GDC computes it.
Sign convention: money TO you = positive (PV for a loan); money FROM you = negative (PMT for repayments).
PMT@ END for amortisation (repayments at the end of each period).

The TVM solver — what each variable means

Fill every known cell, leave the unknown blank, press solve. The GDC handles all the compounding maths internally.

Tip: write out every value you put into the GDC in your working — examiners can’t award method marks for “I used the TVM solver” alone.

🧭 Recipe — any amortisation problem

Identify what’s known: loan amount, rate, repayment, time, compounding frequency.
Set up the TVM solver: PV positive (loan to you), PMT negative (you pay out), FV = 0 (loan ends paid off).
Match P/Y and C/Y to the repayment frequency (usually both 12 for monthly).
Set PMT@ = END for amortisation (repay at end of each period).
Leave blank what you want, solve, then convert (e.g. months → years & months).

Worked examples

WE 1

Find N — how long to pay off

James takes a $50 000 student loan at 5% nominal annual interest, compounded monthly. He repays $400 each month. How long, in years and months, will it take to pay it off?

TVM solver inputs N = ?, I% = 5, PV = 50 000 PMT = −400, FV = 0 P/Y = 12, C/Y = 12, PMT@ = END GDC returns N = 176.94 months Convert to years and months 176.94 ÷ 12 = 14.745 years 0.745 × 12 ≈ 9 months 14 years and 9 months always round UP the months because at month 176 he hasn’t quite finished — he needs that 177th payment to clear it.

WE 2

Find PMT — monthly mortgage payment

Aria takes a £200 000 mortgage at 4.5% nominal annual interest, compounded monthly, over 25 years. Find her monthly repayment.

TVM solver inputs N = 300 (25 × 12), I% = 4.5 PV = 200 000, PMT = ? FV = 0, P/Y = C/Y = 12, PMT@ = END GDC returns PMT = −1111.66 monthly repayment = £1111.66 GDC shows it negative (money out) — quote the answer as a positive amount in the final answer.

WE 3

Total amount paid & interest

For Aria’s mortgage in WE 2, find (a) the total amount she will pay back over 25 years and (b) the total interest paid.

(a) Total = N × PMT total = 300 × 1111.66 = £333 499.49 (b) Interest = total − loan interest = 333 499.49 − 200 000 total paid: £333 499.49, interest: £133 499.49 she’ll pay back over 1.6× the original loan — the cost of borrowing for 25 years.

WE 4

Find PV — how much can you borrow?

You can afford repayments of $1200 per month for 30 years on a mortgage at 6.5% nominal annual, compounded monthly. Find the maximum amount you can borrow.

TVM solver inputs N = 360 (30 × 12), I% = 6.5 PV = ?, PMT = −1200 FV = 0, P/Y = C/Y = 12, PMT@ = END GDC returns PV = 189 852.98 maximum loan ≈ $189 853 useful when budgeting — work backwards from what you can afford each month.

WE 5

Interest on the WE 4 mortgage

Find the total interest paid over the 30 years for the mortgage in WE 4.

Total paid $1200 × 360 = $432 000 Interest = total − loan interest = 432 000 − 189 853 total interest ≈ $242 147 interest exceeds the loan amount itself — that’s what 30 years of compounding does.

WE 6

Compare two interest rates

Compare monthly repayments on a $250 000 mortgage over 20 years for two rates: (a) 3% per annum and (b) 6% per annum, both compounded monthly.

(a) 3% rate N = 240, I% = 3, PV = 250 000 PMT_A = $1386.49/month (b) 6% rate N = 240, I% = 6, PV = 250 000 PMT_B = $1791.08/month Difference 1791.08 − 1386.49 = $404.59/month higher at 6% 6% rate adds ~$405/month, or ~$97 100 over 20 years a 3% point rate increase doubles the cost of borrowing in this case — why mortgage rates matter so much.

💡 Top tips

Write every TVM input on your paper — examiners need to see your method.
Sign rule: money TO you = +, money OUT = −. PV positive for loans, PMT negative.
FV = 0 when the loan is fully paid off.
Round months UP: 176.94 → 177 payments (last one finishes the loan).

⚠ Common mistakes

Wrong sign on PMT: making PMT positive instead of negative gives a nonsense answer (or no answer).
P/Y ≠ C/Y: for AI SL these are always the same. If a question says monthly repayments, both are 12.
Mixing N as years instead of months: 25 years monthly = 300 periods, not 25.
Showing GDC answer with the wrong sign: the calculator displays PMT as negative — your final answer should be positive (an amount paid).

Up next: Annuities — the flip side of amortisation. Instead of paying off a loan, you RECEIVE regular payments from an investment. Same TVM solver, just opposite signs.

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