The big idea: Compound interest pays interest on the interest — so the balance grows geometrically.
If interest is added once a year, FV = PV(1 + r/100)ⁿ.
For example, $1000 at 5% a year for 3 years: 1000 × 1.05³ = $1157.63.
- the future value
- the present value (amount invested)
- the annual interest rate (as a percent)
- compounding periods per year
- the number of years
Compound vs simple: Simple interest adds the same amount every year (arithmetic).
Compound interest multiplies by (1 + rate) each period (geometric) — so it grows faster, and faster still the more often it compounds. This is the geometric idea of [[Growth & decay]] (1.3.3) taken further.
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Split the year into k periods: If interest compounds k times a year (half-yearly k = 2, quarterly k = 4, monthly k = 12), split the year into k periods.
Three steps
Rate per period
per-period rate = r ÷ k.
Number of periods
number of periods = k × n.
Future value
FV = PV(1 + r/(100k))kn.
Why? Each period multiplies by (1 + r/(100k)); over k × n periods that's the power.
IB-style question — quarterly
$5000 is invested at 4% per year, compounded quarterly.
Find the value after 3 years.
Step by step
- Quarterly → k = 4. Per-period rate = 4 ÷ 4 = 1%; periods = 4 × 3 = 12.
- Simplify inside the bracket.
- Work it out.
Final answer
≈ $5634.13.
More frequent = a little more: For the same nominal rate, monthly beats quarterly beats annual — the interest starts earning interest sooner.
The difference is usually small but it earns marks.
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The method depends on the paper: The calculator on Paper 2, the binomial theorem by hand on Paper 1.
Paper 2
- The GDC / TVM solver does it for you (see GDC finance solver, 1.4.3).
- That is the usual way.
Paper 1 (no calculator)
- After one year a quarterly investment is PV(1 + x)⁴, where x is the per-quarter rate.
- Expand the bracket with the binomial theorem and add the terms.
- You learn the binomial theorem itself in 1.9; here you just plug into the expansion you're given.
IB-style question — compound interest by hand
$2000 is invested at 8% per year, compounded quarterly.
(a) Write the amount after one year as 2000(1 + x)⁴ and state x.
(b) Using (1 + x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴, find the amount after one year, to the nearest dollar.
Step by step
- (a) x is the per-quarter rate — the annual rate split over the 4 quarters.
- (b) Substitute x = 0.02 into the given expansion.
- Multiply by the amount invested.
Final answer
x = 0.02; amount ≈ $2165.
Why it's a Paper-1 question: The binomial expansion lets you compute compound growth without a calculator.
Substitute the small per-period rate and the higher-power terms become tiny.