Loans, Annuities and Savings (AI SL)
A loan is repaid, or a savings account built up, through a stream of equal payments made at regular intervals rather than a single lump sum. The IB syllabus doesn't ask you to memorise a formula for this - it's a GDC finance-solver skill, part of the wider Financial Maths topic.
12 questions on this sub-topic.
The GDC finance solver
Covered under IB syllabus reference SL1.7 - amortisation and annuities using technology, whether the built-in financial package of a GDC or a spreadsheet. Payments are made at the end of each period, and knowledge of the annuity formula itself is not required.
TVM solver variables
N, I%, PV, PMT, FV, P/Y, C/Y
N = number of payments, I% = annual interest rate, PV = present value, PMT = the regular payment, FV = future value, P/Y and C/Y = payments and compounding periods per year. Enter every known value, leave the unknown blank, and solve for it.
Not examinable as a formula - solved on the GDC onlyMoney paid out is entered as negative, money received as positive - a loan you take out is a positive PV with a negative repayment PMT; a savings deposit is a negative PMT building up a positive FV. Full keystrokes for each calculator model are on the parent page's GDC guidance.
Worked examples
A $10,000 loan is repaid monthly over 3 years at 6% p.a. compounded monthly.
Find the monthly payment.
Worked solution
Compounded monthly means we work per month. Monthly rate \(i = \frac{6\%}{12} = 0.5\% = 0.005\); number of payments \(n = 3 \times 12 = 36\). A1
The payment that clears a loan \(P\) is
\(PMT = \frac{P\,i}{1 - (1+i)^{-n}} = \frac{10000(0.005)}{1 - (1.005)^{-36}}.\)M1 A1
Denominator \(= 1 - 0.83564 = 0.16436\), so \(PMT = \dfrac{50}{0.16436} \approx $304.22\) A1 per month.
$150 is deposited at the end of each month into an account at 5.4% p.a. compounded monthly.
Find the value after 4 years.
Worked solution
Monthly rate \(i = \frac{0.054}{12} = 0.0045\); deposits \(n = 4 \times 12 = 48\). Deposits are made at the end of each month (an ordinary annuity). A1
\(FV = PMT\cdot\frac{(1+i)^n - 1}{i} = 150\cdot\frac{(1.0045)^{48} - 1}{0.0045}.\)M1
\((1.0045)^{48} = 1.24050\), so \(FV = 150 \times \dfrac{0.24050}{0.0045}\) A1
\(\approx $8016.71.\) A1
Common mistakes
- Leaving the annual rate un-converted. Every N, I%, P/Y and C/Y entry must agree with the payment frequency - a "6% p.a. compounded monthly" loan needs N and P/Y in months, not years, or the solver returns the wrong payment entirely.
- Confusing loan and savings sign conventions. For a loan, PV is positive (you receive it) and PMT comes back negative (you pay it out); for a savings annuity, PV is usually 0 and PMT is entered negative (you're paying in) to build a positive FV.
- Trying to derive the annuity formula from scratch. The syllabus explicitly doesn't require it - reaching for the GDC's finance solver first is both faster and the intended method, so don't waste time reconstructing \(PMT = \tfrac{Pi}{1-(1+i)^{-n}}\) by hand unless a question specifically shows the formula.
Ready to practise properly?
12 loans, annuities and savings questions, marked instantly like the real exam.
Quick answers
Do I need to memorise a formula for loans and annuities?
No. The IB syllabus explicitly does not require the annuity formula itself - these questions are solved with the GDC's built-in finance (TVM) solver, entering N, I%, PV, PMT, FV, P/Y and C/Y.
Why does PV or PMT come out negative on the finance solver?
The solver treats money paid out as negative and money received as positive. If you enter a loan received as a positive PV, the repayments (PMT) return negative - it's just cash-flow direction, not an error.