Compound Interest & Growth (AI HL)
Compound interest is the engine behind almost every financial question on the AI HL paper: a balance that earns interest on interest, growing faster the longer it's left alone. This page narrows in on the growth formula itself and the compounding-frequency traps that catch out most students, as part of the wider Financial Maths topic.
30 questions on this sub-topic.
The two ideas you need
Covered under IB syllabus reference SL1.4: financial applications of geometric sequences and series, including compound interest calculated yearly, half-yearly, quarterly or monthly. Deriving the formula is never examined.
Compound interest / growth
\[FV = PV\left(1+\dfrac{r}{100}\right)^n\]
In the formula booklet. \(PV\) is the amount invested, \(r\) the interest rate per period as a percentage, \(n\) the number of periods.
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 \(\div12\). This isn't in the formula booklet - it's prior knowledge you're expected to apply.
See Financial Maths for GDC finance-solver keystrokes, and how "real value" questions adjust the future value for inflation.
Worked examples
$5000 is invested at 3% p.a. compounded monthly.
Find the value after 4 years, to the nearest cent.
Worked solution
\(A = 5000\left(1 + \dfrac{0.03}{12}\right)^{48} = 5000(1.0025)^{48}.\) M1
\((1.0025)^{48} \approx 1.12733\), so \(A \approx $5636.64.\) A1
On the GDC's finance solver: N=48, I%=3, PV=-5000, PMT=0, P/Y=C/Y=12, then solve for FV.
An investment of $3000 grows to $4000 in 5 years with annual compounding.
Find the annual interest rate, to 2 significant figures.
Worked solution
\(4000 = 3000(1+r)^5 \Rightarrow (1+r)^5\) M1 \(= \tfrac{4}{3}.\) A1
\(1+r = \left(\tfrac43\right)^{1/5}\) M1 \(\approx 1.05922.\) A1
\(r \approx 0.0592 = 5.9\%.\) A1
On the GDC's finance solver: N=5, PV=-3000, FV=4000, P/Y=C/Y=1, then solve for I%.
Common mistakes
- 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.
- 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.
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30 compound-interest and growth questions, marked instantly like the real exam.
Quick answers
What is the compound interest formula for IB AI HL?
\(FV = PV\left(1+\tfrac{r}{100}\right)^n\), where \(PV\) is the amount invested, \(r\) is the interest rate per period as a percentage, and \(n\) is the number of periods.
How do I handle interest that compounds monthly or quarterly instead of yearly?
Divide the annual rate by the number of periods per year to get the rate per period, and multiply the number of years by that same number to get \(n\). For \(4.8\%\) p.a. compounded monthly over 5 years, use a rate of \(4.8/12\) and \(n=60\).