Question

What is the key difference between simple interest and compound interest, and how does this difference affect the effectiveness of each?

Answer 1

Key difference
- Simple interest is computed only on the original principal. Interest does not earn interest.
- Compound interest is computed on the principal plus any interest already earned (interest on interest).

Formulas
- Simple: A = P(1 + r t) where P = principal, r = annual rate, t = years.
- Compound (n compounding periods per year): A = P(1 + r/n)^(n t). For annual compounding n = 1 gives A = P(1 + r)^t. Continuous compounding: A = P e^{r t}.

How this affects effectiveness
- Growth shape: simple interest grows linearly with time; compound interest grows exponentially. That means compound interest produces increasingly larger gains the longer money is left to grow (or, for borrowers, increasingly larger costs).
- Magnitude: for the same P, r and t, compound interest always yields equal or greater amount than simple interest; the advantage grows with higher r, more compounding periods, and longer t.
- Effective annual rate (EAR): with simple interest EAR = r. With compound interest EAR = (1 + r/n)^n − 1 (or e^{r} − 1 for continuous compounding), which is ≥ r and increases with n.

Quick example
- P = $1,000, r = 5%, t = 3 years:
- Simple: A = 1000(1 + 0.05·3) = $1,150
- Annual compound: A = 1000(1.05)^3 ≈ $1,157.63

Bottom line
- For savers/investors, compound interest is generally far more effective at growing wealth over time. For borrowers, compound interest increases the total cost compared with simple interest.