Financial Maths (AI SL)
Financial Maths is geometric sequences applied to money: interest that compounds is just a geometric sequence with a common ratio just above 1. This topic covers simple and compound interest, depreciation, adjusting an investment's value for inflation, and the amortization of loans and annuities using your GDC's built-in finance solver. It's the most calculator-dependent topic on the syllabus - the IB deliberately doesn't expect you to derive these formulas, only to set the right inputs and interpret the output.
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. Examination questions may require the use of technology, including built-in financial packages. |
| SL1.7 | Amortization and annuities using technology (the built-in financial packages of a GDC or a spreadsheet). Payments are made at the end of each period. Knowledge of the annuity formula itself is not required - the GDC's finance solver is used instead. |
Both sit in the Number & Algebra unit and build directly on the geometric sequence formulas from SL1.3.
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 principal - it never earns interest on interest already paid, so it grows in a straight line over time.
e.g. \($2000\) at 5% p.a. simple interest for 4 years: \(I = 2000(0.05)(4) = $400\).
What is compound interest?
Compound interest adds each period's interest back onto the balance, so future interest is calculated on a growing total. This makes it a geometric sequence with common ratio \(1+\tfrac{r}{100}\).
e.g. \($2000\) at 5% p.a. compounded annually for 4 years: \(2000(1.05)^4 \approx $2431.01\).
What is depreciation?
Depreciation is the loss in value of an asset over time, often modelled as a fixed percentage decrease each year - the same geometric structure as compound interest, but with a ratio below 1.
e.g. a \($10\,000\) machine depreciating 10% p.a.: after 3 years it's worth \(10\,000(0.9)^3 = $7290\).
What is real value?
Real value is an investment's nominal future value adjusted for inflation, expressed in today's purchasing power. It's found by dividing the nominal value by the inflation growth factor.
e.g. a nominal value of \($12\,155.06\) after 4 years of 3% inflation is worth \(12\,155.06 \div (1.03)^4 \approx $10\,800\) today.
What is amortization?
Amortization is paying off a loan through a series of regular payments, each covering that period's interest plus a portion of the remaining principal, until the balance reaches zero.
e.g. an \($18\,000\) loan at 9% compounded monthly over 5 years amortizes with a monthly repayment of \($373.65\).
Key formulas
Financial Maths is unusual in that the GDC does most of the algebraic work - your job is choosing the right formula or the right finance-solver inputs. The tables below cover both.
Formula reference
Simple and compound interest are in the formula booklet; the annuity formula behind amortization is deliberately not examinable - the syllabus states you use the GDC's finance solver instead.
| Formula | Used for | Booklet? |
|---|---|---|
| \(I = Prt\) | Simple interest | ✓ Yes |
| \(FV = PV\left(1+\dfrac{r}{100k}\right)^{kn}\) | Compound interest / depreciation (\(k\) = compounding periods per year) | ✓ Yes |
| real value \(= \dfrac{\text{nominal value}}{(1+i)^n}\) | Adjusting for inflation rate \(i\) | Not in the formula booklet - derived from the compound interest formula |
| N, I%, PV, PMT, FV, P/Y, C/Y | Amortization and annuities (GDC finance solver) | Not examinable as a formula - solved on the GDC only |
Simple vs compound interest
Both start from the same principal and rate, but only one lets interest earn further interest - which matters more the longer the money is invested.
| Feature | Simple interest | Compound interest |
|---|---|---|
| Formula | \(I = Prt\) | \(FV = PV(1+\tfrac{r}{100})^n\) |
| Interest calculated on | Original principal only | Principal plus all interest so far |
| Growth shape | Straight line | Exponential curve |
| \($2000\) at 5% for 4 years | \($400\) interest | \($431.01\) interest |
Compound growth and decay
The same formula models growth (interest) or decay (depreciation) - only the sign of the rate changes.
Annual compounding
\[FV = PV(1+\tfrac{r}{100})^n\]
Interest is added once per year, so \(n\) is simply the number of years.
✓ In the formula bookletNon-annual compounding
\[FV = PV\left(1+\dfrac{r}{100k}\right)^{kn}\]
Divide the annual rate by \(k\) periods, and multiply the years by \(k\) - e.g. monthly means \(k=12\).
✓ In the formula bookletReal value and inflation
\[\text{real value} = \dfrac{\text{nominal value}}{(1+i)^n}\]
The nominal (headline) gain overstates the true increase in purchasing power once inflation is accounted for.
Not in the formula booklet - derived techniqueLoans and annuities
Amortization spreads a loan or savings goal into regular payments - this is entirely GDC-driven, using the finance solver's six standard inputs.
Finance solver inputs
N (number of payments), I% (annual rate), PV, PMT, FV, and P/Y = C/Y (payments and compounds per year).
Money you pay out is negative, money you receive is positive - sign errors are the most common mistake here.
Not examinable as a formula - GDC onlyAmortizing a loan
Enter the loan as a positive PV, set FV = 0, and solve for PMT to find the regular repayment that clears the debt.
Total repaid \(=\) PMT \(\times\) N; total interest \(=\) total repaid \(-\) loan amount.
Not examinable as a formula - GDC onlyBuilding an annuity
Enter regular deposits as a negative PMT with PV = 0, and solve for FV to find the value built up over time.
Interest earned \(=\) FV \(-\) (PMT \(\times\) number of payments).
Not examinable as a formula - GDC onlyWorked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
$2000 is invested for 4 years at 5% p.a.
(a) Find the simple interest earned.
(b) Find the value with annual compounding.
(c) Find the difference.
Worked solution
(a) Simple interest. \(I = Prt = 2000(0.05)(4) = $400.\) A1 Simple interest is a fixed amount each year on the original principal.
(b) Compound value. \(A = 2000(1.05)^4 = 2000(1.21551)\) M1 \(\approx $2431.01.\) A1
(c) Difference. Compound interest earned \(= 2431.01 - 2000 = $431.01\); difference over simple \(= 431.01 - 400\) M1
\(\approx $31.01.\) A1 Compounding earns interest-on-interest, so it always beats simple interest over more than one period.
A family wants $30,000 for a deposit. They invest $22,000 now at 4.5% p.a. compounded annually.
(a) Find the value after 5 years.
(b) Show this is not yet $30,000.
(c) Find the least whole number of years to reach $30,000.
Worked solution
(a) Value after 5 years. \(22000(1.045)^5 = 22000(1.24618)\) M1 \(\approx $27\,415.\) A1
(b) Not yet enough. \($27\,415 < $30\,000\), so 5 years falls short of the deposit goal. R1
(c) Years to reach $30 000. \(22000(1.045)^n = 30000 \Rightarrow (1.045)^n = 1.3636 \Rightarrow n = \frac{\ln 1.3636}{\ln 1.045} \approx 7.04.\)M1 A1
The target is passed during the 8th year, so round up: \(n = 8\) years. R1 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 \(I=Prt\) for compound growth. Simple interest formulas only apply when there's genuinely no interest-on-interest - anything described as "compounded" needs \(FV=PV(1+\tfrac{r}{100})^n\) instead.
- Sign errors in the finance solver. Money paid out (an investment, a loan repayment) should be negative; money received (a loan amount, a maturity value) should be positive. Mixing the signs gives an answer with the wrong sign or magnitude.
- Forgetting to convert the rate and time for non-annual compounding. Monthly compounding needs the annual rate divided by 12 and the number of years multiplied by 12 - using the annual rate directly overstates the growth.
- Reading the nominal value as the real value. A headline gain of \($2155\) sounds large, but once divided by the inflation factor the true increase in purchasing power is smaller - always divide by \((1+i)^n\) if inflation is mentioned.
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.
Faster and safer than algebra for "how many years until..." questions, where the unknown \(n\) sits inside a compound-growth exponent.
- 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: For loans and annuities specifically, your GDC's dedicated finance solver (TVM Solver on TI-84, Finance Solver on Nspire, the Financial app on Casio) is usually faster than a general numerical solver - enter N, I%, PV, PMT, FV, P/Y and C/Y and let it solve for the missing one.
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 simple and compound interest?
Simple interest pays a fixed amount each year on the original principal only, so growth is a straight line: \(I = Prt\). Compound interest pays interest on the interest already earned, so growth accelerates: \(FV = PV\left(1+\dfrac{r}{100}\right)^n\). Over more than one period, compound interest always earns more.
How do I handle interest compounded monthly instead of annually?
Divide the annual rate by the number of compounding periods per year, and multiply the number of years by that same number of periods. For 4.5% compounded monthly over 6 years: \(FV = PV\left(1+\dfrac{0.045}{12}\right)^{72}\).
How do I find the real value of an investment after inflation?
Divide the nominal (face-value) future amount by \((1+i)^n\), where \(i\) is the inflation rate and \(n\) is the number of years. This converts the amount into today's purchasing power, which is usually smaller than the headline nominal figure.
Can I use my GDC for this topic?
Yes, and you're expected to. The built-in finance solver (TVM Solver on TI-84, Finance Solver on Nspire, the Financial app on Casio) handles compound interest, loans, and annuities directly - you enter the known values and it solves for the one you're missing. 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 SL syllabus unit, in case you want to keep going.