The magic of Compounding and the Rule of 70
Source: Chapter 24 By: Mankiw
It may be tempting to dismiss differences in growth rates as insignificant. If one country grows at 1 percent while another grows at 3 percent, so what? What difference can 2 percent make?
The answer is: a big difference. Even growth rates that seem small when written in percentage terms seem large after they are compounded for many years. Compounding refers to the accumulation of a growth rate over a period of time.
Consider an example. Suppose that two college graduates -- Jerry and Elaine -- both take their first jobs at the age of 22 earning $30,000 a year. Jerry lives in an economy where all incomes grow at 1 percent per year, while Elaine lives in one where incomes grow at 3 percent per year. Straightforward calculations show what happens. Forty years later, when both are 62 years old, Jerry earns $45,000 a year, while Elaine earns $98,000. Because of that difference of 2 percentage points in the growth rate, Elaine's salary is more than twice Jerry's.
An old rule of thumb, called the rule of 70, is helpful in understanding growth rates and the effects of compounding. According to the rule of 70, if some variable grows at a rate of x percent per year, then that variable doubles in approximately 70/x years. In Jerry's economy, incomes grow at 1 percent per year, so it takes about 70 years for incomes to double. In Elaine's economy, incomes grow at 3 percent per year, so it takes about 70/3, or 23, years for incomes to double.
The rule of 70 applies not only to growing economy but also to a growing savings account. Here is an example: In 1791, Ben Franklin died and left $5,000 to be invested for a period of 200 years to benefit medical students and scientific research. If this money had earned 7 percent per year (which would, in fact, have been very possible to do), the investment would have doubled in value every 10 years. Over 200 years, it would have doubled 20 times. At the end of 200 years of compounding, the investment would have been worth (2的20次方 * $5,000), which is about $5 billion. (In fact, Franklin's $5,000 grew to only $2 billion over 200 years because some of the money was spent along the way.)
As these examples show, growth rates compounded over many years can lead to some spectacular results. That is probably why Albert Einstein once called compounding "the greatest mathematical discovery of all time."
It may be tempting to dismiss differences in growth rates as insignificant. If one country grows at 1 percent while another grows at 3 percent, so what? What difference can 2 percent make?
The answer is: a big difference. Even growth rates that seem small when written in percentage terms seem large after they are compounded for many years. Compounding refers to the accumulation of a growth rate over a period of time.
Consider an example. Suppose that two college graduates -- Jerry and Elaine -- both take their first jobs at the age of 22 earning $30,000 a year. Jerry lives in an economy where all incomes grow at 1 percent per year, while Elaine lives in one where incomes grow at 3 percent per year. Straightforward calculations show what happens. Forty years later, when both are 62 years old, Jerry earns $45,000 a year, while Elaine earns $98,000. Because of that difference of 2 percentage points in the growth rate, Elaine's salary is more than twice Jerry's.
An old rule of thumb, called the rule of 70, is helpful in understanding growth rates and the effects of compounding. According to the rule of 70, if some variable grows at a rate of x percent per year, then that variable doubles in approximately 70/x years. In Jerry's economy, incomes grow at 1 percent per year, so it takes about 70 years for incomes to double. In Elaine's economy, incomes grow at 3 percent per year, so it takes about 70/3, or 23, years for incomes to double.
The rule of 70 applies not only to growing economy but also to a growing savings account. Here is an example: In 1791, Ben Franklin died and left $5,000 to be invested for a period of 200 years to benefit medical students and scientific research. If this money had earned 7 percent per year (which would, in fact, have been very possible to do), the investment would have doubled in value every 10 years. Over 200 years, it would have doubled 20 times. At the end of 200 years of compounding, the investment would have been worth (2的20次方 * $5,000), which is about $5 billion. (In fact, Franklin's $5,000 grew to only $2 billion over 200 years because some of the money was spent along the way.)
As these examples show, growth rates compounded over many years can lead to some spectacular results. That is probably why Albert Einstein once called compounding "the greatest mathematical discovery of all time."
posted by tickup | 4:44 PM
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