What Is Compound Interest? Simple vs. Compound Explained with Formulas, Tables, and the Rule of 72

The difference between simple and compound interest

With simple interest, interest is paid only on the original principal. With compound interest, the interest earned is added to the principal, and the next interest is calculated on “principal + interest so far.” The first year is the same either way, but from the second year on, compound interest is calculated on a larger base.

• Simple interest total: principal × (1 + annual rate × years)
• Compound interest total (compounded once a year): principal × (1 + annual rate)^years
• Compound interest total (compounded m times a year): principal × (1 + annual rate ÷ m)^(m × years)

Leaving ¥1,000,000 at 5% for 10 years, simple interest gives 1,000,000 × (1 + 0.05 × 10) = ¥1,500,000. Annual compounding gives 1,000,000 × 1.05^10 ≈ ¥1,628,895, about ¥129,000 more. That ¥129,000 is the total “interest on interest.”

The gap widens over time

Here are the totals for ¥1,000,000 invested at 5% with simple interest and with compound interest (compounded annually), by number of years.

Simple interest is a straight line that grows by the same ¥50,000 each year, while compound interest is a curve whose growth itself gets bigger every year. After 5 years the difference is only about ¥26,000, but after 30 years it is about ¥1,820,000, nearly twice the original principal. This is why people say the most important factor in compounding is not the rate of return but time.

By rate of return, ¥1,000,000 grows in 30 years to about ¥1,060,000 at 0.2% (a typical Japanese time-deposit rate), about ¥1,350,000 at 1%, about ¥2,430,000 at 3%, about ¥4,320,000 at 5%, and about ¥7,610,000 at 7%. Over long periods, small differences in rate turn into differences of several times.

Compounding frequency and the effective annual rate

Even at the same “5% a year,” the more often interest is added to the principal, the slightly larger the actual growth over a year (the effective annual rate). Comparing ¥1,000,000 at 5% for 10 years:

The difference between annual and daily compounding over 10 years is about ¥20,000. Compounding frequency matters far less than the rate or the number of years, so when comparing products, looking at the stated annual rate is usually enough. Note that the timing of compounding varies by product, such as semiannual compounding on some time deposits or interest paid every six months on some government bonds.

Use the Rule of 72 to find how long money takes to double

The Rule of 72 is the rule of thumb “72 ÷ annual rate (%) ≈ years for money to double.” It lets you estimate the time in your head without a calculator, which is handy for getting a feel for rates of return. It matches the exact figure for annual compounding closely.

It works in reverse too. To double your money in 10 years, you need 72 ÷ 10 = 7.2% a year. In fact, ¥1,000,000 at 7.2% for 10 years becomes about ¥2,000,000, and after 30 years about ¥8,050,000 (doubling three times, roughly eight-fold). On the other hand, a deposit at 0.2% would take 360 years to double.

How not to lose the compounding effect

• Don’t withdraw gains along the way: taking distributions or spending profits every time you lock them in removes the interest-on-interest part and brings you closer to simple interest. For mutual funds, choosing the type that reinvests distributions (accumulation funds) is the basic choice.
• Defer taxes or avoid them: in a taxable account, tax is taken each time you realize a gain (20.315% in Japan), shrinking the base for compounding. Tax-advantaged accounts such as Japan’s NISA (or ISAs and IRAs elsewhere) prevent this erosion.
• Keep costs low: fund management fees are deducted every year, which has the same effect as lowering your return by that amount.
• Stay invested for longer: as the table shows, the compounding gap grows larger in the later years. Holding for a long time benefits more than frequent short-term trading.

Compounding works the same way when you invest a fixed amount every month. Future values by monthly amount and number of years are in the monthly investing simulation tables, and the difference made by tax-free accounts is in new NISA vs. taxable accounts.

Compare compound and simple interest with the tool

1. 1

   Open the “Lump sum & compound” tab

   In the “Lump sum & compound” tab of the Savings & Compound Interest Simulator, enter the principal (the lump sum), the expected rate of return, and the period.

2. 2

   Choose the compounding frequency

   Choosing annual, semiannual, quarterly, monthly, or daily switches the compound amount and the effective annual rate.

3. 3

   See the gap from simple interest and the years to double

   The compound and simple totals, the difference between them, and the years for the principal to double (both the exact value and the Rule of 72 estimate) are shown side by side, and a chart shows the two lines spreading apart.

FAQ

Which is better, compound or simple interest?

If you are the one growing money (savings or investments), compound interest is better. ¥1,000,000 at 5% for 10 years becomes ¥1,500,000 with simple interest and about ¥1,630,000 with annual compounding. If you are the borrower, compounding makes interest balloon and the burden heavier.

What is the compound interest formula?

With annual compounding, it’s “principal × (1 + annual rate)^years.” Compounded m times a year, it’s “principal × (1 + annual rate ÷ m)^(m × years).” For ¥1,000,000 at 5% for 10 years, 1,000,000 × 1.05^10 ≈ ¥1,628,895.

What is the Rule of 72?

It’s a rule of thumb: divide 72 by the annual rate (%) to get the approximate number of years for money to double. At 6% it doubles in 12 years, at 3% in 24 years. It closely matches the exact figures for annual compounding (11.9 years at 6%, 23.4 years at 3%), and it also works in reverse, as in “to double in 10 years you need 7.2% a year.”

How much difference is there between monthly and annual compounding?

At 5% a year, the effective annual rate is 5.000% with annual compounding and 5.116% with monthly compounding. On ¥1,000,000 over 10 years, the difference is about ¥18,000, a small effect compared with differences in rate or time.

Do bank deposits use compound interest?

In Japan, time deposits come in simple-interest and semiannual-compounding types, and semiannual compounding is sometimes available for deposits of three years or more. However, Japanese deposit rates are often around 0.1% to 0.3% a year, and ¥1,000,000 left for 30 years at 0.2% only becomes about ¥1,060,000, so the effect of compounding is barely noticeable. Check how your own bank compounds and what rate it pays.