Rule of 72

What is Rule of 72?

The rule of 72 meaning refers to a simple mathematical formula used to estimate the time it takes for an investment to double at a given interest rate. This rule is based on the mathematical principle of compounding interest and is particularly useful for making quick approximations without the need for complex calculations. 

By dividing 72 by the annual rate of return, the Rule of 72 provides investors with a rough estimation of the number of years it takes for their investment to grow twofold. The simplicity of this concept makes it an invaluable tool for investors and financial planners alike, allowing for a general understanding of an investment's potential growth and helping in the decision-making process.
 

About Rule 72?

By dividing the number 72 by the annual rate of return (expressed as a percentage), the Rule of 72 provides a rough estimation of the number of years needed for an investment to grow twofold. Although not an exact method, this rule offers a reasonably accurate approximation, especially for interest rates between 6% and 10%. It is important to note that the Rule of 72 assumes a constant rate of return and uninterrupted compounding.
The practicality of the Rule of 72 lies in its ability to simplify complex financial calculations, thus making it easier to compare and evaluate different investment options. By providing a general sense of how long it will take for an investment to double, it assists investors in making informed decisions and planning for future financial goals. Moreover, the Rule of 72 can also be applied to other areas, such as understanding the impact of inflation on purchasing power or gauging the effects of economic growth.
 

How can you use the Rule of 72?

To use this rule, simply divide 72 by the annual rate of return (expressed as a percentage), and the resulting number will give you an approximation of the number of years needed for the investment to grow twofold.
For instance, if you are considering investing in a financial instrument that offers an 8% annual rate of return, dividing 72 by 8 will yield 9 years as the approximate time needed to double your investment. This mental shortcut allows you to easily compare different investment opportunities and make informed decisions about where to allocate your resources.
The Rule of 72 can be applied in other financial contexts as well. For instance, you can use it to understand the impact of inflation on your money's purchasing power. By dividing 72 by the annual inflation rate, you can estimate the number of years it will take for the purchasing power of your money to be halved.

Time (Years) to Double an Investment

Here’s a table that compares the results obtained using the Rule of 72 with the actual time it takes for an investment to double, based on different annual rates of return:

This table demonstrates that the Rule of 72 provides a close approximation of the actual time needed for an investment to double at various rates of return. While not 100% accurate, the rule offers a quick and easy way to make estimates that are generally reliable for financial planning purposes.

Rule of 72 Formula

The Rule of 72 is a formula used to estimate the number of periods required to double an investment's value, based on a given interest rate per period. The formula is as follows:

t = 72 / r
Where, 
t = number of periods required to double an investment's value
 r = interest rate per period, as a percentage

Example of the Rule of 72

Here’s a rule of 72 example: 

The Rule of 72 can be illustrated by taking an investment with a fixed annual interest rate of 8%. Dividing 72 by the interest rate of 8 gives the number of years required to double the investment, which in this case is 9 years. So, if an individual invests Rs. 10,000 at an 8% annual interest rate, they can expect the investment to double to Rs. 20,000 in approximately nine years. The Rule of 72 provides a quick and easy way to estimate the number of years required to double an investment based on the given interest rate.
 

Deriving the Rule of 72

The Rule of 72 is a quick and simple method to estimate the time required to double an investment's value. It is derived from the compound interest formula, which is used to calculate the future value of an investment over a period of time. The formula for compound interest is:

A = P (1 + r/n)^(nt)

Where: A = Future value of the investment P = Principal amount r = Interest rate n = Number of times the interest is compounded per year t = Time in years

To derive the Rule of 72, we can simplify the above formula by assuming that the interest rate and compounding period are constant. In this case, the formula becomes:

A = P (1 + r)^t

By applying natural logarithm to both sides, we obtain:

ln A = ln P + t ln (1 + r)

Using the first-order Taylor series approximation, we can approximate ln (1 + r) as r. Therefore, the formula becomes:

ln A ≈ ln P + r t

Rearranging this equation, we get:

t ≈ ln 2 / r

This is the Rule of 72, where t represents the number of years required to double an investment's value, and r is the interest rate per year.
 

Rules of 72, 69.3, and 69

The Rule of 72, Rule of 69.3, and Rule of 69 are all formulas used to approximate the number of years it will take for an investment to double in value based on a given interest rate. 

●    The Rule of 72 states that by dividing 72 by the annual interest rate, you can estimate the number of years required for an investment to double. 
●    The Rule of 69.3 is a more accurate formula for higher interest rates and is calculated by dividing 69.3 by the interest rate. 
●    The Rule of 69 is another approximation formula used for continuous compounding and is calculated by dividing 69 by the interest rate. 

These rules are helpful tools for quickly estimating the potential growth of an investment.
 

Advantages and Disadvantages of Rule of 72

The Rule of 72 is a quick and easy way to estimate the time required to double an investment. It has several advantages, such as being easy to understand, apply and calculate, making it a useful tool for investors. It can be used to compare different investment options and help investors make informed decisions about where to put their money.
However, the Rule of 72 is based on a few assumptions that may not always be accurate, such as a constant rate of return and compounding period. It also does not take into account taxes, inflation, and other factors that may impact investment returns. Therefore, it is important to use the Rule of 72 as a rough estimate and not rely solely on it for making investment decisions.
 

Rule of 72 vs 70

The difference between the two is that the Rule of 72 uses the number 72, while the Rule of 70 uses the number 70. The Rule of 70, which is based on the natural logarithm of 2, is a more precise formula with an approximate value of 0.693.
This means that the Rule of 70 will produce more precise results when calculating the time it takes for an investment to double. However, the Rule of 72 is simpler to use and provides a close approximation that is easy to remember.