Compound Interest
Compound interest occurs when interest earned on an initial investment or deposit begins earning interest itself, creating exponential growth rather than linear growth. The formula A = P(1 + r)^n calculates the final amount, where P represents principal, r the annual interest rate as a decimal, and n the number of years. This mathematical concept forms the foundation of long-term wealth building and appears in savings accounts, investment funds, and retirement planning.
Why it matters
Compound interest drives the mathematics behind retirement savings, college funds, and long-term investments. A person who invests $5,000 annually starting at age 25 will accumulate approximately $1,142,000 by age 65 at a 7% return, while someone starting at age 35 with the same contributions reaches only $540,000. This $602,000 difference illustrates why financial advisors emphasize early investing. The concept appears in mortgage calculations, where borrowers pay compound interest to lenders, and in credit card debt, where unpaid balances compound monthly. Index funds rely on compound growth to build wealth over decades, turning modest monthly contributions into substantial retirement accounts through the mathematical power of exponential growth.
How to solve compound interest
Compound Interest
- Compound interest earns interest on both the original principal AND on previously earned interest — that's why the curve bends upward over time.
- Annual: A = P(1 + r)n, where P is the principal, r the rate as a decimal, n the number of years.
- Monthly compounding: A = P(1 + r/12)12n.
- With monthly contributions PMT: future value = PMT × [((1 + i)n − 1) / i], where i = r/12 and n is the number of months.
- Index funds and savings accounts both rely on this — small early differences in rate, time, or starting age compound to outsized differences at the end.
Example: $10,000 at 7% for 20 years: A = 10000 · 1.07²⁰ ≈ $38,697.
Worked examples
You deposit $7,500.00 in a savings account paying 5% interest per year. How much is in the account after one year?
Answer: 7875
- Calculate the interest → 7500 × 5100 = 375 — Interest = principal × rate. $7,500.00 × 5% = $375.00.
- Add the interest to the principal → 7500 + 375 = 7875 — After one year: $7,500.00 + $375.00 = $7,875.00.
You invest $8,000.00 at 3% compound interest per year. What is the value after 3 years?
Answer: 8742
- Use the compound interest formula → A = P(1 + r)n — P is the principal, r the rate as a decimal, and n the number of years.
- Plug in the values → A = 8000 × (1 + 0.03)3 — P = $8,000.00, r = 0.03, n = 3.
- Compute the growth factor → (1 + 0.03)3 = 1.0927 — Raise 1 + r to the power n.
- Multiply by the principal → A ≈ $8,742.00 — $8,000.00 × 1.0927 ≈ $8,742.00 after rounding.
You invest $15,000.00 in an index fund returning 5% per year. What is the value after 5 years, assuming returns are reinvested?
Answer: 19144
- Apply A = P(1 + r)n → A = 15000(1 + 0.05)5 — Reinvested returns compound — the formula treats each year's gain as next year's principal.
- Compute the result → A ≈ $19,144.00 — After 5 years, $15,000.00 grows to approximately $19,144.00 — a gain of $4,144.00.
Common mistakes
- Confusing simple and compound interest calculations — computing $1,000 at 5% for 3 years as $1,150 (simple interest) instead of $1,157.63 (compound interest).
- Forgetting to convert percentage rates to decimals — using 0.05^3 instead of (1.05)^3 when calculating compound growth.
- Mixing up principal and final amount in the formula — writing P = A(1 + r)^n instead of A = P(1 + r)^n.