MONEYSCHOOL Financial Education for Real Life!

> The Rule of 72
> Downside Math
> Compounding Power
> Millions & Billions: Getting A Sense of Large Numbers

The Rule of 72
This easy mental math trick allows you to estimate the number of years it takes for money to double when it's invested at a certain interest rate.

Here's how it works:
Divide 72 by the interest rate and you get the approximate number of years needed to double your money when it's invested at that interest rate.

Example:
Let's say you put $1,000 dollars in a savings account that pays interest at 6% per year. Divide 72 by 6 and the result is 12.

So at 6% interest, it would take 12 years for savings to double from $1,000 to $2,000, without depositing any more money in the account. If the rate of return is not fixed, you can use the rate you expect to earn on the investment.

Be sure you use the interest rate (6) and not the decimal equivalent (0.06) when you do this calculation. And, remember, the Rule of 72 works only if you let your interest or earnings accumulate in the account.

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Downside Math
There’s always some risk when you invest money. In general, to earn higher returns, you have to take on more risk. But smart investors know it’s vital to protect yourself from significant losses. Many investments move both up and down, but a little math shows why the ups and downs are not equally important.

Let’s say you have $1,000 invested in a stock. If the stock price rises 10%, your investment has increased in value to $1,100. If it falls 10%, your investment is worth only $900. Here’s the problem. Once you have lost part of your principal, you are working from a smaller base. So to make up for a 10% loss, you need to gain over 11% on your remaining money (1.11111 X 900 = 1000). The more you lose, the worse it gets, as shown in the following examples.

Examples:
If you lose 20% of $1000, you have $800 left, so it takes a 25% gain to get even
(1.25 X $800 = $1000).

If you lose one-third of $1000, you have $666.67 left, so it takes a 50% gain to get even
(1.5 X $666.67 = $1000).

If you lose 50% of $1000, you have $500 left, so you have to double your money to get even
(2 X $500 = $1000).

These examples show why it is so important to pay attention to the downside risk on your investments.

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Compounding Power
One of the surest ways to get rich is to save money and invest it over many years. What makes that strategy work is the power of compound interest. In case you’ve forgotten, compounding takes place when the interest you earn on your money is added to the principal over and over, so that interest is paid not only on the original principal, but also on all of the interest accumulated in the account over time.

The exact calculation depends on how often compounding takes place. Depending on the rules that apply to a particular type of account, interest could be compounded annually, quarterly, monthly, daily, or continuously. The more frequently the money is compounded, the more interest you earn on the account.

Illustration:
Here’s an illustration that will open your eyes to the power of compounding. Suppose someone offered you a choice of: (1) An outright gift of $1 million; or, (2) a penny deposited in a savings account that doubles every day for 30 days.

As you may have guessed, this is a trick question, but the only trick is that when interest is compounded on such a high interest rate (in this case, 100% per day), even a tiny initial amount grows into an immense sum very quickly.

The savings account that started with a penny will grow to $10,737,418.24 in 30 days. You can check the calculation yourself. Just get out your calculator. Or you can do the calculation in a spreadsheet program. In Microsoft Excel, the formula would be =0.01*2^30.

Note: We have seen other versions of this illustration, but this one came from p. 87 of The Complete Idiot’s Guide to Managing Your Money, by Robert K. Heady and Christy Heady, with Hugo Ottolenghi (Fourth Edition, Alpha Books, 2005). We did correct one error. In the book, the final amount is shown as $5.37 million, but that’s the amount you get if a penny is doubled 29 times.

That would be the correct result if you deposited your money in an account for 30 days, and it doubled each day. Because the first doubling would take a whole day, the result would not show up until the second day, and you would have only 29 doublings in 30 days. But the way the choice is stated above, the money “doubles every day for 30 days,” so we think the correct amount is the one we give above.

Either way, this fun illustration makes the point that compound interests is very powerful. But you also need to know that in real life, a return of 10% per year is more realistic, and at that rate, accumulating wealth takes a lot longer.

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Millions and Billions:
Getting A Sense of Large Numbers
To make sense of the financial world, we have to deal with numbers, and many people just aren’t very comfortable with math. In his book Innumeracy: Mathematical Illiteracy and Its Consequences, Temple University Professor John Allen Paulos attempts to straighten out some common misconceptions about numbers. The book was on the New York Times best-seller list for 18 weeks after it was originally published in 1989. A new edition was published in 2005, and Professor Paulos has written other books on the same subject. The problems he writes about are still with us, and his books are certainly worth reading.

Princeton Professor Brian Kernighan teaches a course on Quantitative Reasoning, in which he also attempts to help students learn “numeric self defense.” Among other examples, he shows many cases where respected newspapers and magazines confuse “millions,” “billions,” and “trillions.”

Examples:
Here is an example Paulos uses in Innumeracy to help non-mathematicians get a sense of large numbers. Paulos points out that a million seconds is about 11.5 days, while a billion seconds is almost 32 years (31.71 to be exact). A trillion seconds then, would be 31,710 years.

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