Compound Interest Explained

Simple and Compound Interest

With simple interest, only the money you put in earns interest. 10,000 at 5% earns 500 every year, so after 10 years you have 15,000. With compound interest, each year's interest is added to the balance and earns interest itself: the second year pays 5% on 10,500, the third on 11,025, and after 10 years you have 16,288.95.

The Formula

A single deposit P at an annual rate r, compounded n times a year for t years, grows to

A = P × (1 + r ÷ n)^{n × t}

With yearly compounding n = 1; with monthly compounding n = 12, so each month adds r ÷ 12 of the balance. Simple interest is A = P × (1 + r × t).

How Much Compounding Frequency Matters

Compounding more often adds a little, because interest starts earning interest sooner. The effective annual rate, often called APY (annual percentage yield), puts the effect into one figure: (1 + r ÷ n)ⁿ − 1.

The jump from yearly to monthly compounding is worth less than raising the rate by a quarter of a point. When comparing savings accounts, compare their effective annual rates rather than the nominal ones.

Regular Deposits

Most people save a fixed amount every month rather than one lump sum. Each deposit then grows for a different length of time. With deposits D made at the end of each month at a monthly rate i, the balance after m months is D × ((1 + i)^m − 1) ÷ i. Paying at the start of each month instead multiplies that by (1 + i), because every deposit earns one more month.

For example, a starting amount of 10,000 plus 200 a month for 10 years (120 payments) at 5% compounded monthly, with payment 1 made on opening, reaches 47,655.95. Of that, 34,000 is money paid in and 13,655.95 is interest. The calculator shows the same figures and a year-by-year table under "Lump sum + deposits". If the monthly payments start one month after opening instead, the total is 47,526.55.

The Rule of 72

Dividing 72 by the annual rate gives roughly the years needed to double your money: about 10 years at 7.2%, 14.4 years at 5%, 24 years at 3%. The rule works best for rates between about 4% and 12%. The exact doubling time is ln 2 ÷ (n × ln(1 + r ÷ n)), which the calculator shows for the rate and compounding you choose: 14.2 years at 5% compounded yearly.

What the Calculation Leaves Out

• Tax on interest or investment gains, which reduces the effective rate.
• Inflation: money that doubles in 14 years buys less than twice as much.
• Changing rates and, for investments, returns that vary from year to year and can be negative.
• Fees, and banks that credit interest only on set dates.

Frequently Asked Questions