# The Power of Compound Interest

## The Magic of the “Rule of 72” February 27, 2014

Compound interest is a subtle, yet powerful way to increase your earnings — and when comparing compound interest rates available to you, the "Rule of 72" is an extraordinarily simple way to determine the effect on your savings and investments.

Let's illustrate with a simple example of compound interest. You have \$100 invested at 10% interest. At the end of the first compounding period, your \$100 will have earned \$10 of interest, for \$110 total.

That \$110 will earn \$11 in interest in the next period; think of that as \$10 earned by your original principal and \$1 earned by your interest from the first period. For the next period, you have \$121, which will subsequently earn \$12.10 — \$10 from the original principal, \$1.10 from the first block of earned interest, and \$1 from the second block of earned interest. This rate of growth is exponential.

So what is the "Rule of 72"? It’s a shortcut to figure out how long it will take for your money to double at a given interest rate. (If you are allergic to math, you may want to skip this part.) To find the inverse of an exponential process, you'll need to use the natural logarithm (ln) function. Applied to periodic compounding of interest, the formula is:

t = ln(x)/ln(1 + r)

where r is the interest rate as a decimal, t is number of compounding periods, and x is the factor by which your money will increase. For low to medium interest rates, the denominator ln(1 + r) is virtually the same as the interest rate itself. (At 5% interest, or 0.05, ln(1.05) = 0.0488.)

For your money to double, the numerator of the equation becomes ln(2) — which happens to be close to 0.72. It's actually 0.693, but 0.72 is preferred for a few reasons:

• It’s a quick calculation;
• 72 is divisible by so many other numbers;
• The denominator isn’t exactly the same as the interest rate, so it’s OK to adjust the numerator;
• Finally, “The Rule of 69.3” just doesn't sound as snappy.

Math class is almost over, except for the rule itself:

Time = 72/interest rate (as a whole number)

At 6% annual interest, your principal will double in 12 years. At 8% annual interest, it will double in 9 years. At 10%, as in the example above, your money will double in a little over 7 years.

It's easy to see the effect of compounding periods with this rule. For example, 1.5% compounded quarterly doubles your money in 48 quarterly periods — or 12 years, the same as 6% annually! (In reality, the former is slightly better than the latter.) This makes the Rule of 72 really useful for thumbnail comparisons of different compounding rates and periods.

Since we are comparing relative changes in principal, the formula works regardless of the principal amount — \$100 or \$100 million. (The error does increase with higher interest rates.)

The Rule of 72 also works in other ways:

• Rate Estimates – Swap time and interest rate in the above equation and you can solve the alternate problem: what interest rate do I need to double my money in a particular amount of time?

• Estimating Losses – How long would it take for your capital to degrade by half, or at what interest rate would it degrade by half over a given time?

• Buying power – At a given rate of annual inflation, how long would it take the buying power of your assets to decrease by half?

• Effect of Fees – The same principle as above, taking regular fees out of an account.

The Rule of 72 is good for other financial estimates such as economic growth or decline, or estimating college tuition costs. Feel free to use it the next time you are comparing rates for your investment opportunities. Remember, the rule works well for anything where the underlying changes are exponential in nature, and you're comparing relative changes.

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