Loans, Annuities and Savings (AA SL)
A single lump sum growing at compound interest is only half the picture - many exam questions instead involve a series of regular deposits or repayments, each one growing for a different length of time. This page covers how to set those questions up on the GDC's finance solver, and the timing detail (start vs end of period) that most often costs marks. It's part of the broader Financial Maths topic.
11 questions on this sub-topic.
Ordinary annuities vs annuities due
Covered under IB syllabus reference SL1.4: financial applications of geometric sequences and series, extended here to regular deposits and loan repayments. Neither formula below is in the formula booklet - these questions are designed to be solved with the GDC's built-in finance (TVM) solver, not by hand.
Ordinary annuity (end of period)
\[FV=PMT\cdot\dfrac{(1+i)^{n}-1}{i}\]
Deposits or payments land at the end of each period. \(i\) is the interest rate per period and \(n\) is the number of periods. On the GDC, this is PMT:END mode.
Annuity due (start of period)
\[FV_{\text{due}}=FV_{\text{ordinary}}\times(1+i)\]
Deposits land at the start of each period, so every one earns an extra period of interest. On the GDC, this is PMT:BEGIN mode.
Need the compound interest and depreciation formulas, or a full GDC finance-solver walkthrough? See Financial Maths.
Worked examples
Kofi deposits $150 at the end of every month into a savings account paying a nominal annual interest rate of 4.2%, compounded monthly.
Find the value of the account immediately after the 36th deposit.
Worked solution
This is an ordinary annuity (deposits at the end of each period): \(N=36,\ I\%=4.2,\ PMT=-150,\ P/Y=C/Y=12,\ PV=0.\) M1
\(i=\dfrac{0.042}{12}=0.0035.\) A1
\(FV=150\cdot\dfrac{(1.0035)^{36}-1}{0.0035}.\) M1
\(FV\approx $5744.26.\) A1
At the start of every quarter, Amara deposits $500 into an account that pays a nominal annual interest rate of 5%, compounded quarterly.
Find the value of the account immediately after the 12th deposit (3 years).
Worked solution
Deposits occur at the start of each quarter, so this is an annuity due (BEGIN mode): \(N=12,\ I\%=5,\ PMT=-500,\ P/Y=C/Y=4.\) M1
\(i=\dfrac{0.05}{4}=0.0125.\) A1
\(FV_{\text{ordinary}}=500\cdot\dfrac{(1.0125)^{12}-1}{0.0125},\qquad FV_{\text{due}}\) M1 \(=FV_{\text{ordinary}}\times 1.0125.\) A1
\(FV_{\text{due}}\approx $6510.56.\) A1
Common mistakes
- Missing the start-vs-end timing. A question that says deposits happen "at the start of" each period needs BEGIN mode (or the extra \(\times(1+i)\) factor) - using the plain ordinary-annuity formula undercounts every deposit by one period of growth.
- Applying the nominal annual rate directly. A "5% compounded quarterly" account needs the rate divided by 4 and the number of periods multiplied by 4 before anything is substituted - the nominal rate itself is never used as \(i\) in the formula.
- Confusing N with the number of years. \(N\) on the GDC is the total number of payment periods, not years - for monthly deposits over 3 years, \(N=36\), not \(3\).
Ready to practise properly?
11 loan, annuity and savings questions, marked instantly like the real exam.
Quick answers
What is the difference between an ordinary annuity and an annuity due?
In an ordinary annuity, deposits or payments happen at the end of each period (END mode on the GDC). In an annuity due, they happen at the start of each period (BEGIN mode), so every deposit earns one extra period of interest and the final value is the ordinary value multiplied by \((1+i)\).
Do I need to memorise an annuity formula for the exam?
No - annuity questions are designed to be solved on the GDC's built-in finance (TVM) solver by entering N, I%, PV, PMT, P/Y and C/Y, not by substituting into a formula by hand.