Financial Maths (AA SL)
Money that earns interest grows geometrically, not linearly - which is why financial maths sits inside the sequences and series unit. This topic covers simple interest as a stepping stone into compound interest, calculating the value of an investment or loan after any number of compounding periods, and modelling depreciation as a geometric decrease, almost always using your GDC's built-in financial solver.
What the syllabus says
This topic maps onto one point in the official IB Analysis & Approaches syllabus.
| Code | Syllabus content |
|---|---|
| SL1.4 | Financial applications of geometric sequences and series: compound interest and annual depreciation, including calculating the real value of an investment given an interest rate and an inflation rate. Examination questions may require the use of technology, including built-in financial packages. The concept of simple interest may be used as an introduction to compound interest. Compound interest can be calculated yearly, half-yearly, quarterly or monthly. Questions that ask students to derive the formula will not be set. |
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is simple interest?
Simple interest pays a fixed amount each period, calculated only on the original amount invested (the principal) - it never earns interest on interest already paid. It grows the total value in a straight line over time.
e.g. \($2000\) at \(5\%\) simple interest for \(3\) years earns \(2000\times0.05\times3=$300\).
What is compound interest?
Compound interest pays interest on the principal plus all interest already earned, so the total grows geometrically rather than in a straight line. Each compounding period multiplies the current value by a fixed growth factor.
e.g. \($2000\) at \(4\%\) compounded annually for \(1\) year grows to \(2000(1.04)=$2080\).
What is the compounding frequency?
The compounding frequency is how often interest is added each year - yearly, half-yearly, quarterly or monthly. A higher frequency splits the annual rate into more (smaller) periods, but the interest itself compounds more often, so the total return is slightly higher.
e.g. A nominal \(6\%\) rate compounded monthly uses a monthly rate of \(6\div12=0.5\%\) per period.
What is depreciation?
Depreciation is a decrease in value over time, most commonly modelled as a fixed percentage reduction each year - a geometric sequence with a ratio less than \(1\). It's used for assets like vehicles and machinery that lose value with age.
e.g. A \($24\,000\) machine depreciating \(15\%\) per year is worth \(24\,000(0.85)=$20\,400\) after 1 year.
What is the TVM (time value of money) solver?
The TVM solver is a built-in GDC tool for financial calculations - you enter values like the number of periods, interest rate, present value and payment, and it solves for whichever one is missing, such as the future value or the payment amount.
e.g. \(N=5,\ I\%=4,\ PV=-2000,\ P/Y=C/Y=1\) solves for \(FV\approx2433.31\).
Key formulas
Financial maths is a geometric sequence in disguise - the compound interest formula is just \(u_n=u_1r^{n-1}\) written with financial variable names.
Formula reference
The compound interest formula is given in the formula booklet in its financial form; simple interest and depreciation follow directly from prior-knowledge percentage rules.
| Formula | Used for | Booklet? |
|---|---|---|
| \(I=Prt\) | Simple interest | Not in booklet - prior knowledge |
| \(FV=PV\left(1+\dfrac{r}{100k}\right)^{kn}\) | Compound interest (value after \(n\) years, \(k\) compounds/year) | ✓ Yes |
| \(V=V_0(1-r)^{n}\) | Depreciation (value after \(n\) years) | Not in booklet - same structure as compound interest with a negative rate |
Simple vs compound interest
Both start from the same principal and rate, but compound interest overtakes simple interest as soon as more than one period has passed.
| Feature | Simple interest | Compound interest |
|---|---|---|
| Interest earned each period | Same fixed amount | Grows each period |
| Calculated on | Original principal only | Principal plus all interest so far |
| Growth shape | Linear | Geometric (exponential) |
| Formula | \(I=Prt\) | \(FV=PV(1+i)^{n}\) |
| Example (\($4000\), \(6\%\), 3 yrs) | \($4720\) | \($4764.06\) |
Compound interest
The compounding frequency changes how the nominal annual rate splits into periods.
Annual compounding
\[FV=PV(1+i)^{n}\]
One compound per year - \(i\) is the annual rate and \(n\) is the number of years.
✓ In the formula bookletMonthly compounding
\[FV=PV\left(1+\dfrac{r}{1200}\right)^{12n}\]
Divide the nominal annual rate by 12 for the monthly rate, and multiply the years by 12 for the number of periods.
✓ In the formula bookletSolving for time
To find how long an investment takes to reach a target, set up the inequality and solve using logarithms or the GDC's numerical solver - not by trial and error.
Depreciation and loans
Depreciation is compound interest with a ratio below 1; loans use the same TVM structure in reverse.
Depreciation
\[V=V_0(1-r)^{n}\]
A geometric sequence with common ratio \(1-r\), where \(r\) is the annual depreciation rate.
Loan repayments
Loans compound in the same way as savings, but the TVM solver's payment (PMT) function finds the regular instalment needed to clear the balance over a fixed term.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
\($4000\) is invested for 3 years at 6% per year.
(a) Find the interest under simple interest.
(b) Find the value under compound interest (annual).
(c) State which earns more.
Worked solution
(a) Simple interest. \(I=Prt=4000(0.06)(3)=$720\), value \(=$4720.\) M1 A1
(b) Compound. \(4000(1.06)^3\) M1
\(\approx$4764.06.\) A1
(c) Compound earns more since \($4764.06>$4720.\) A1
Maria saves money over 12 months. Plan A: she saves \($50\) in month 1 and increases the amount by \($10\) each month. Plan B: she saves \($30\) in month 1 and increases the amount by 8% each month.
(a) For Plan A, find the amount saved in month 12.
(b) For Plan A, find the total saved over the 12 months.
(c) For Plan B, find the total saved over the 12 months, to the nearest dollar.
(d) State which plan saves more in total, and by how much.
Worked solution
(a) Plan A is arithmetic, \(u_1=50,\ d=10\): \(u_{12}=u_1+11d=50+110\) M1
\(=$160.\) A1
(b) \(S_{12}=\tfrac{12}{2}(u_1+u_{12})=6(210)\) M1
\(=$1260.\) A1
(c) Plan B is geometric, \(u_1=30,\ r=1.08\): \(S_{12}=\dfrac{30(1.08^{12}-1)}{0.08}\) M1
\(\approx$569.\) A1
(d) Plan A saves more by \(1260-569\) M1
\(=$691.\) A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Using the annual rate directly for monthly compounding. A nominal \(6\%\) rate compounded monthly needs a monthly rate of \(0.5\%\) and \(12\) periods per year - not \(6\%\) applied twelve times.
- Confusing simple and compound interest. Simple interest is linear (\(I=Prt\)); compound interest is geometric (\(FV=PV(1+i)^n\)) - mixing the two gives the wrong growth pattern entirely.
- Forgetting depreciation uses a ratio below 1. A machine depreciating \(15\%\) per year has common ratio \(1-0.15=0.85\), not \(0.15\) itself.
- Rounding too early in multi-step problems. Carry full calculator accuracy through each step and only round the final answer - rounding an intermediate value can shift a "least whole year" answer by a full year.
Using your GDC
The step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
Faster and safer than algebra for messy equations - useful for finding the least number of years an investment or loan takes to cross a target value.
- Graph \(f(n)\) first so you can see how many solutions exist and roughly where they are.
- Rearrange so everything is on one side: \(f(n) = 0\) - or graph both sides as separate functions and find intersections.
- MATH → Solver: enter the expression, type a starting guess close to one root, press ALPHA + ENTER. Move the guess to near a different root and repeat for each solution.TI-84
- Type nSolve(f(n)=0, n, guess) - include a guess or interval e.g. nSolve(f(n)=0, n, 2) or nSolve(f(n)=0, n, {1,5}) to target a specific root.Nspire
- Run-Matrix → SolveN(f(n), n) returns all real roots at once; or use the Equation app for a visual approach.Casio
- Always verify each solution by substituting back into the original equation.
Tip: The solver finds ONE root near your starting guess - change the guess to find others. The graph shows you how many to expect.
See the full GDC guide for more calculator models and topics, including your model's built-in financial (TVM) solver.
Ready to practise properly?
Financial maths questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What's the difference between simple and compound interest?
Simple interest earns a fixed amount each period, calculated only on the original principal. Compound interest earns interest on the principal plus all previously earned interest, so the amount grows faster over time - which is why compound interest is modelled as a geometric sequence.
How often can compound interest be compounded?
Yearly, half-yearly, quarterly or monthly are all common on the IB syllabus. The more often interest compounds within a year, the more total interest is earned, even at the same nominal annual rate - because each compounding period earns interest on interest already added.
Do I need to derive the compound interest formula?
No - IB exam questions will not ask you to derive the compound interest formula. You're expected to apply it (or your GDC's financial solver) correctly, not prove where it comes from.
Can I use my GDC for financial maths questions?
Yes - and you're expected to. Financial maths questions on Paper 2 are designed around your GDC's built-in financial (TVM) solver or equation solver, since the exponents involved are rarely convenient to work out by hand. See the GDC guide for model-specific instructions.
Sub-topics
Financial Maths broken down into its individual skills, each with its own focused page.
Related topics
More Number & Algebra topics from the same AA SL syllabus unit, in case you want to keep going.