Compound Interest
What this domain tests
Compound interest earns interest on previously earned interest, so the balance grows geometrically. Items ask for the compound amount, the interest earned, the effective annual rate, or a comparison against simple interest.
Core rules
- Compound amount: A = P(1 + i)^n. Exponentiate first, then multiply by the principal.
- Period rate: i = annual rate ÷ number of compounding periods per year (m). Annual 8% compounded quarterly gives i = 0.02.
- Number of periods: n = years × m.
- Compound interest = A − P.
- Effective annual rate: EAR = (1 + r/m)^m − 1. It is the true yearly growth rate and is always at least the nominal rate; the two are equal only when compounding is annual.
- Continuous compounding: A = Pe^(rt).
- Common m values: annually 1, semiannually 2, quarterly 4, monthly 12, daily 360 or 365.
- Discounting: P = A ÷ (1 + i)^n finds the present value of a future amount.
- Rule of 72: 72 ÷ annual rate approximates the years needed to double.
Worked example
$2,000 is deposited at 6% compounded quarterly for 2 years. Find the compound amount and the interest earned. i = 0.06 ÷ 4 = 0.015; n = 2 × 4 = 8. A = 2,000(1.015)^8 = 2,000 × 1.126493 = $2,252.99. Interest = 2,252.99 − 2,000 = $252.99. Simple interest for the same term would be 2,000 × 0.06 × 2 = $240, so compounding added $12.99.
Traps that cost marks
- Using the annual rate with n left in years while compounding is quarterly or monthly.
- Converting only half the formula: dividing the rate by 4 but leaving n in years.
- Reporting the compound amount when the question asks for interest, or the reverse.
- Assuming the nominal rate equals the effective rate.
- Rounding (1 + i)^n too early, which shifts the answer by dollars.
Where it appears
Savings accounts, certificates of deposit, credit-card balances, and reinvested bond coupons are all quoted with a nominal rate plus a compounding frequency, which is exactly the information these items supply.