Financial Maths (AI HL)
Financial maths applies geometric sequences to money - how an investment, loan or asset changes value over time. This topic covers compound interest, depreciation, the real value of an investment once inflation is taken into account, and amortization and annuities, where regular payments pay off a loan or build up savings. It's the most calculator-dependent topic in the syllabus: your GDC's finance solver does the heavy lifting.
What the syllabus says
This topic maps onto two points in the official IB Applications & Interpretation syllabus.
| Code | Syllabus content |
|---|---|
| SL1.4 | Financial applications of geometric sequences and series: compound interest and annual depreciation. Calculating the real value of an investment given an interest rate and an inflation rate. Compound interest may be calculated yearly, half-yearly, quarterly or monthly. Deriving the formula will not be examined. |
| SL1.7 | Amortization and annuities using technology, including built-in financial packages on a GDC or spreadsheet. In examinations, payments are made at the end of each period. Knowledge of the annuity formula will enhance understanding but will not be examined. |
Both are core AI content, examined at both SL and HL.
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 compound interest?
Compound interest is interest calculated on the current balance, not just the original amount - so each period's interest is added to the balance before the next period's interest is worked out. The value after \(n\) periods is \(FV = PV\left(1+\dfrac{r}{100}\right)^n\).
e.g. \($1000\) at \(5\%\) p.a. for 3 years: \(FV=1000(1.05)^3=$1157.63\).
What is depreciation?
Depreciation is the loss in value of an asset over time - it uses the same compound formula as interest, but with a negative rate, since the value shrinks by a fixed percentage each period instead of growing.
e.g. A \($20\,000\) machine depreciating at \(15\%\) p.a. for 4 years: \(20000(0.85)^4\approx$10\,440.13\).
What is the real value of an investment?
The real value adjusts a nominal (face-value) amount for inflation, showing what it can actually buy. You divide the nominal future value by \((1+\text{inflation rate})^n\) to strip out the effect of rising prices.
e.g. \($12\,000\) nominal after 5 years with \(2\%\) p.a. inflation has real value \(12000/(1.02)^5\approx$10\,868.76\).
What is an annuity?
An annuity is a series of equal, regular payments - either paying off a loan (amortization) or building up savings. The IB doesn't require you to derive the annuity formula; you use your GDC's finance app to find the unknown payment, rate or time.
e.g. A \($6000\) loan at \(5\%\) p.a. repaid in 3 equal annual instalments needs a payment of about \($2203.25\) per year, found via the finance solver.
What is compounding frequency?
Compounding frequency is how often interest is added to the balance within a year - annually, half-yearly, quarterly or monthly. More frequent compounding gives a slightly higher final value for the same nominal annual rate, since interest starts earning interest sooner.
e.g. \(6\%\) p.a. compounded monthly on \($1\) for a year: \((1+0.06/12)^{12}\approx1.0617\), an effective rate of about \(6.17\%\).
Key formulas
One core formula, reused for growth, decay and adjusted for inflation - plus a finance solver for anything involving regular payments. The tables below summarise all of it at a glance.
Formula reference
The compound interest formula is on the official formula booklet. The annuity/amortization formula is deliberately not examinable - the syllabus states it will not be required, so the GDC finance app does that work instead.
| Formula | Used for | Booklet? |
|---|---|---|
| \(FV = PV\left(1+\dfrac{r}{100}\right)^n\) | Compound interest / growth | ✓ Yes |
| \(FV = PV\left(1-\dfrac{r}{100}\right)^n\) | Depreciation (same formula, negative rate) | Not listed separately - same formula as above |
| Real value \(=\dfrac{\text{nominal value}}{(1+i)^n}\) | Adjusting for inflation rate \(i\) | Not in the formula booklet - prior knowledge |
| Annuity / amortization formula | Regular payments on a loan or savings plan | Not examined - use the GDC finance solver |
Growth vs decay
Compound interest (growth) and depreciation (decay) are the same mathematical structure viewed from opposite directions - only the sign of the rate changes.
| Feature | Compound interest | Depreciation |
|---|---|---|
| Formula | \(FV=PV(1+\tfrac{r}{100})^n\) | \(FV=PV(1-\tfrac{r}{100})^n\) |
| Direction | Value increases | Value decreases |
| Typical example | Savings account, bond | Car, machinery, equipment |
| Long-run behaviour | Grows without bound | Approaches (but never reaches) zero |
Compound interest & depreciation
Both use the same one formula from the formula booklet - the only choice you make is the sign of the rate, and how many compounding periods there are per year.
Compounding periods
\[n = (\text{years})\times(\text{periods per year})\]
For monthly compounding over 5 years, \(n=60\) and the rate per period is the annual rate \(\div12\).
Not in the formula booklet - prior knowledgeDepreciation
\[FV = PV\left(1-\tfrac{r}{100}\right)^n\]
Same structure as compound interest, but the rate subtracts from 1 instead of adding to it.
Not listed separately - use the compound interest formulaInterest earned
\[\text{Interest} = FV - PV\]
The total interest earned (or value lost, for depreciation) is just the difference between the final and initial values.
Not in the formula booklet - prior knowledgeAmortization, annuities & real value
These questions are recognisable because they involve regular payments or an inflation-adjusted comparison, rather than a single lump sum.
Amortization
Paying off a loan through equal regular instalments, each of which covers that period's interest plus some of the principal. Solved with the GDC's finance (TVM) solver, not by hand.
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 technologyReal value
\[\text{Real value} = \dfrac{FV_{\text{nominal}}}{(1+i)^n}\]
Strips out the effect of inflation \(i\) so you can compare purchasing power, not just face value, across time.
Not in the formula booklet - prior knowledgeWorked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
\($3500\) is invested at \(4.2\%\) p.a. compounded annually.
(a) Find the value after 5 years, to the nearest cent.
(b) Find the interest earned.
Worked solution
(a) \(A = 3500(1.042)^5\) M1
\((1.042)^5 \approx 1.22840,\ A\approx $4299.39.\) A1
(b) \($799.39.\) A1
\($5000\) is invested at a nominal \(4.8\%\) p.a. for 5 years.
(a) Find the value with annual compounding.
(b) Find the value with monthly compounding.
Give each to the nearest cent.
Worked solution
(a) \(5000(1.048)^5\) M1
\(\approx $6320.86.\) A1
(b) \(5000(1+0.048/12)^{60} = 5000(1.004)^{60}\) M1
\(\approx $6353.20.\) 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.
- Forgetting to convert the rate and time to match the compounding frequency. For monthly compounding, the rate per period is the annual rate \(\div12\) and \(n\) is the number of months, not years - mixing annual and monthly units is the single biggest source of lost marks.
- Using annual figures when the question specifies a different frequency. A nominal \(4.8\%\) p.a. compounded monthly is not the same as \(4.8\%\) compounded annually - always re-read which frequency the question asks for before setting up the formula.
- Confusing real value with nominal value. A "real value" question wants the inflation-adjusted figure, dividing by \((1+i)^n\) - reporting the plain compound-interest future value instead loses the whole point of the question.
- Sign errors in the GDC's TVM solver. Most finance apps require PV and FV/PMT to have opposite signs (money paid out is negative, money received is positive) - getting this backwards flips the sign of your final answer.
Using your GDC
Every 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.
Useful when a question asks you to find an unknown rate or number of years - for example "how long until the investment doubles?" - rather than an unknown final value.
- Graph \(f(x)\) first so you can see how many solutions exist and roughly where they are.
- Rearrange so everything is on one side: \(f(x) = 0\) - 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(x)=0, x, guess) - include a guess or interval e.g. nSolve(f(x)=0, x, 2) or nSolve(f(x)=0, x, {1,5}) to target a specific root.Nspire
- Run-Matrix → SolveN(f(x), x) 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.
Every major GDC also has a dedicated finance (TVM) solver for compound interest, depreciation and amortization problems - you enter the values you know (number of periods, interest rate, present value, payment, future value) and solve for the one you don't. The exact menu path differs by model, so see the full GDC guide for the steps on yours.
See the full GDC guide for more calculator models and topics.
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 compound interest and simple interest?
Simple interest is calculated only on the original principal, so it grows by the same dollar amount every period. Compound interest is calculated on the principal plus any interest already earned, so it grows faster over time - the interest itself starts earning interest.
How does compounding frequency affect the final value?
The more often interest is compounded within a year (monthly instead of annually, for example), the higher the final value, because interest starts earning interest sooner. You divide the annual rate by the number of periods per year and multiply the number of years by that same number of periods.
Do I need to memorize the annuity formula?
No. The IB syllabus explicitly states that knowledge of the annuity formula may help your understanding but questions requiring you to derive or quote it will not be set. You're expected to solve amortization and annuity problems using your GDC's built-in finance solver.
Can I use my GDC for this topic?
Yes, and you're expected to - financial maths is entirely calculator-permitted in AI. Compound interest, depreciation, and especially amortization/annuities are designed around your calculator's finance (TVM) app or a spreadsheet, not hand calculation. 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 AI HL syllabus unit, in case you want to keep going.