What is compound interest (short definition)
– Compound interest is interest calculated on the original amount you put in (the principal) plus on interest that has already been added to that balance. In other words, you earn (or pay) interest on interest. A compounding period is the interval at which accumulated interest is added to the account balance.
Key variables and formulas
– P = principal (starting amount)
– r = nominal annual interest rate (decimal, e.g., 5% = 0.05)
– n = number of compounding periods per year (e.g., 12 for monthly)
– t = time in years
– A = future value (principal + interest)
Standard discrete compounding formula:
– A = P × (1 + r/n)^(n×t)
Interest earned or owed:
– Interest = A − P
Continuous compounding (mathematical ideal):
– A = P × e^(r×t) where e ≈ 2.71828
Worked numeric examples
1) Loan example (annual compounding)
– Problem: 3-year loan, P = $10,000, r = 5% (0.05), compounded annually (n = 1), t = 3.
– Step 1: A = 10000 × (1 + 0.05/1)^(1×3) = 10000 × 1.157625 = $11,576.25
– Step 2: Interest = $11,576.25 − $10,000 = $1,576.25
2) Savings: simple interest vs monthly compound (10 years)
– Simple interest over 10 years: Interest = P × r × t = 100,000 × 0.05 × 10 = $50,000.
– Monthly compound: n = 12, A = 100,000 × (1 + 0.05/12)^(12×10) ≈ $164,701 → Interest ≈ $64,701.
– Takeaway: More frequent compounding increases the total interest earned.
Rule-of-72 (quick estimate)
– Divide 72 by the annual return rate (%) to estimate years to double. Example: 72 ÷ 4% ≈ 18 years.
How compounding frequency matters
– Common frequencies: annually, semiannually, quarterly, monthly, daily, or
continuous (compounded continuously).
How frequency changes outcomes — formulas and intuition
– Discrete compounding (n periods per year): A = P × (1 + r/n)^(n t)
– P = principal (initial amount)
– r = nominal annual rate (decimal)
– n = compounding frequency per year (1, 2, 4, 12, 365, …)
– t = time in years
– Effective annual rate (EAR) — the actual annual growth rate after compounding once per year:
– EAR = (1 + r/n)^n − 1
– EAR lets you compare offers with different compounding schedules.
– Continuous compounding (the limiting case as n → ∞): A = P × e^(r t)
– e is Euler’s number ≈ 2.718281828…
– EAR (continuous) = e^r − 1
Worked numeric example (one-year horizon, r = 5%, P = $1,000)
Step 1 — compute EAR for several frequencies:
– Annual