simple vs compound interest

Suppose you put 100 dollars in an account that earns 10 percent a year. There are two honest ways the interest could be calculated, and they part company after the first year. Under simple interest, you earn 10 dollars every year on the original 100 only, so after three years you have 130 dollars. Under compound interest, you earn interest on your interest too: the 10 dollars earned in year one itself starts earning, so after three years you have about 133.10 dollars. The gap looks tiny at first but grows relentlessly over long horizons.

Simple interest grows the balance in a straight line: the accumulated value after n years at rate i is the principal times (1 + i times n). Compound interest grows it as a curve that bends upward: the accumulated value is the principal times (1 + i) raised to the power n. With compounding, each period's interest is added to the balance and then earns interest in every later period — this is the snowball effect. Over one period the two methods agree exactly; only over multiple periods does compounding pull ahead.

In actuarial work, compound interest is the standard and the assumption behind almost every formula in interest theory, because it reflects how investments actually behave when earnings are reinvested. Simple interest survives mostly in short-term contexts — some money-market quotes, certain late-payment penalties, and day-count conventions for instruments under a year. A common error among beginners is to use simple interest over many years; it quietly and substantially understates how money really grows.