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# What is rule 72?

Definition

The Rule of 72 is a formula for estimating the time it will take to double or lose half the value of your investment.

#### What is the rule of 72?

The Rule of 72 is a formula that can help you estimate the effect of exponential growth or exponential decline. This calculation is a simplified version of the original logarithmic formula. The Rule of 72 provides a rough estimate of the time it will take to double or halve an investment without the use of a scientific calculator or logarithmic tables. It’s worth remembering that the Rule of 72 doesn’t take into account any fees or taxes that affect your income if you’re calculating growth.

The formula for calculating the time period for doubling your investment using the Rule of 72 is as follows:

72/ Interest Rate = Years to double

#### History of the Rule of 72

The number 72 was first mentioned in 1494 by Italian mathematician Luca Pacioli in his book “Summa de arithmetica geometria, proportioni et proportionalità” (“Summa de arithmetica geometria, proportioni et proportionalità”). Pacioli made an important point: the number 72 can be used to determine the number of years to double your assets.

The Rule of 72 was written almost a century later. It is based on the standard formula for compound interest: A = P (1 + r/n) nt.

‘A’ is the interest you earned plus your principal (the total of your investments).

‘P’ is the principal or initial investment.

‘r’ is the interest rate in decimal form.

‘n’ is the number of compounding periods.

‘t’ is the time in years.

If we want to double our money, we can substitute A = 2 and P = 1. This leaves us with 2 = 1 ( 1 + r/n) nt.

Assuming that the interest rate increases every year, we can substitute n for 1. We now have 2 = 1 ( 1 + r/1)1*t. We can simplify this equation to 2 = (1 + r)t. Now let’s take the logarithm of both sides to further simplify the equation: ln2 = ln (1 + r )nt.

Then use the rule of degree to reduce the exponent of the degree. ln2 = t * ln (1 + r). The natural logarithm of 2 is approximately 0.693. And for small values of r, ln ( 1 + r ) ≈ r. In other words, we can say that 0.693 ≈ t * r.

We can multiply both parts by 100 to use the interest rate as a whole number rather than a decimal. So we have 69.3 ≈ t * r (where r is the interest rate). Finally, to define t as the number of years it would take to double our investment, we can divide by 100r to get 69.3 / r ≈ t (where r is the interest).

Since 69.3 is a number that is difficult to divide by, statisticians and investors have agreed to use the next closest integer with many divisible multipliers, 72. So 72 divided by the interest rate (expressed as a percentage) gives you the approximate time (number of years) it will take to double your investment.

#### What does the rule of 72 show?

People like to see their money grow – especially as their investments double. Since most people can’t figure out the formula for doubling their assets without a calculator, Rule 72 is a useful inference to give a rough estimate of when an investment will double.

An important difference in this rule is that it does not use simple interest (the amount of the original investment multiplied by the interest rate multiplied by time). Rule 72 uses compound interest (the interest on your initial investment plus the interest earned on your previous interest). In other words, the rule of 72 assumes that every time an investment earns interest, you reinvest it.

#### How do you calculate the number of years using the rule of 72?

The Rule of 72, unlike deriving the 72 formula itself, requires only division, no math. To estimate the doubling time of almost any investment, you need to divide 72 by the annual growth rate. You have to remember to use the whole number of the interest rate in the formula, not a percentage or decimal fraction.

For example, let’s say you have a \$3 investment with a fixed interest rate of 6% per year. 72 divided by 3 equals 26. Thus, it will take 26 years for your \$3 to grow to \$6.

The rule of 72 can also tell you the declining value of an investment. For example, if inflation is 6%, 72 divided by 6 tells you that in about 12 (72/6) years, your money will be worth about half its current value. On the other hand, if inflation drops to 4%, your money will lose half its value in 18 years (72/4).

#### Rule 69 vs. rule 70 vs. rule 72

To calculate the number of years for deposits with annualized interest rates, Rule 72 works best.

Rule 70, on the other hand, is better suited for semi-annual interest accrual. Let’s look at this with an example. Let’s say you have an investment that has an interest rate of 8% and accrues interest semi-annually (or biannually).

Under the rule of 72, you would get 72/8 = 9 years. If you count by rule 70, you get 70/8 = 8.75 years.

Rule 69 gives more accurate results if you calculate continuously (in which case you reinvest the interest continuously and as often as possible), such as monthly or daily.

Consider all three rules for an investment that has an interest rate of 2% per day.

Under rule 72, you will double your money in 36 years (72/2 = 36).

According to rule 70, you will double your money in about 35 years (70/2 = 35).

But rule 69 says you will double your money in 34.5 years (69/2 = 34.5).

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