Loans, Annuities & Savings (AI HL)
An annuity is a stream of equal, regular payments - either paying down a loan (amortization) or building up a savings pot. These questions are never solved by hand: the whole point of this syllabus item is knowing which numbers go into your GDC's finance solver and reading the result correctly. This page sits within the wider Financial Maths topic and focuses only on annuities.
12 questions on this sub-topic.
What the exam expects
Covered under IB syllabus reference SL1.7: amortization and annuities using technology, including a GDC's built-in financial package or a spreadsheet. Payments are always made at the end of each period.
Amortization (loan)
Regular equal payments gradually pay off a lump sum borrowed today, such as a mortgage or car loan. Solved entirely on the GDC's finance solver.
Not examined as a formula - use technologyAnnuity (savings)
Equal regular payments building up towards a future value, such as a retirement fund - the mirror image of amortization, also solved via the finance solver.
Not examined as a formula - use technologyKnowing the underlying annuity formula can help you check your GDC output, but it will never be tested directly. See Financial Maths for full finance-solver keystrokes on each calculator model.
Worked examples
A loan is repaid by 60 monthly payments of $415.17 (from a $20,000 loan).
(a) Find the total amount repaid.
(b) Find the total interest paid.
Worked solution
(a) Total repaid \(= 60 \times 415.17\) M1
\(= $24\,910.20.\) A1
(b) Interest \(= 24\,910.20 - 20\,000\) M1
\(= $4910.20.\) A1
A company must accumulate $100,000 in 8 years by equal monthly deposits into an account earning 4.2% p.a. compounded monthly.
Find the required monthly deposit, to the nearest cent.
Worked solution
\(i = 0.0035,\ n\) M1
\(n = 96.\) A1
\(PMT = \dfrac{100000 \times 0.0035}{1.0035^{96}-1}.\) M1
\(1.0035^{96}\approx1.39992\), so \(PMT = \dfrac{350}{0.39992} \approx $875.18.\) A1
On the GDC's finance solver: N=96, I%=4.2, FV=100000, PV=0, P/Y=C/Y=12, then solve for PMT.
Common mistakes
- Entering the annual rate instead of the period rate. With monthly payments, the finance solver still wants the annual nominal rate in the I% field, but P/Y and C/Y must both be set to 12 - entering an already-divided rate double-counts the conversion.
- Mixing up the sign convention. Money paid out (a deposit, a repayment) and money received (the loan amount, the final balance) need opposite signs on most GDC finance solvers - getting this backwards flips the answer's sign or size.
- Confusing the number of payments with the number of years. \(n\) in the finance solver is the total number of payment periods, not years - for monthly payments over 8 years, that's \(n=96\), not \(n=8\).
Ready to practise properly?
12 loan, annuity and savings questions, marked instantly like the real exam.
Quick answers
Do I need to memorise the annuity formula for the exam?
No. Knowledge of the annuity formula improves understanding, but it's never tested directly - amortization and annuity questions are solved using the GDC's built-in finance (TVM) solver.
What is the difference between a loan (amortization) and a savings annuity?
Both involve regular equal payments, but a loan starts with a lump sum that payments pay off over time, while a savings annuity starts at zero and payments build towards a future lump sum.