Compound Interest

In simple interest the interest is worked out on the original principal every year, so the same amount is added each time. In compound interest the interest of each year is added to the principal, and the next year's interest is calculated on this larger sum. Interest is thus earned on interest. Because the principal keeps growing, the interest in each successive year is bigger than the year before, and the money grows faster than under simple interest.

You could compute compound interest year by year: find one year's interest, add it to the principal, then repeat. This works but becomes tedious for many years. Instead we use a single formula. If a principal P is invested at r per cent per year for n years, the amount becomes A = P(1 + r/100)^n. The compound interest is then CI = A - P. This formula does in one step what the year-by-year method does in many.

Sometimes interest is added twice a year instead of once. Then each half-year carries half the annual rate, and the number of compounding periods doubles. So we use A = P(1 + r/200)^(2n). For example, 10 per cent per year compounded half-yearly is treated as 5 per cent every six months. Because interest is added more often, half-yearly compounding gives a slightly larger amount than yearly compounding at the same annual rate.

The same compounding idea describes things that grow or shrink by a fixed percentage each year. A population or a price that rises by r per cent a year follows Final = Initial(1 + r/100)^n, exactly like compound interest. A machine, car or asset that loses r per cent of its value each year depreciates, following Final = Initial(1 - r/100)^n. The minus sign is the only change: growth multiplies by (1 + r/100), depreciation by (1 - r/100).