## **What is the Rule of 72?**

A basic question that investors often have is, “How long will it take for my investment to double?” This question can be answered quickly and fairly accurately if one knows a simple rule: The Rule of 72. According to the Rule of 72, the number of years it takes to double your investment can be easily estimated by just dividing the number 72 by the annual compounded rate of return. Sounds easy? It really is that easy!

## **Rule of 72 formula**

The Rule of 72 formula is also simple. To calculate the number of years required to double your investment, you use the formula below:

*Number of years required to double investment = 72/compounded rate of return*

Let’s understand the use of the rule with an example.

## **Rule of 72 example**

If you want to find out how much time it will take for $100 to double by investing in an investment product that offers 8% compounded rate of return, all you have to do is one simple calculation.

Number of years required to double investment= 72/8

= 9

Thus, it will take around 9 years for an investment of $100 to become $200. This is a simple example of the investment Rule of 72.

Further, we can also reverse calculate the rate of return required to double your money, if you seek to double your money within a fixed number of years. Let’s understand it with a simple example.

Peter, age 50, wishes to retire at the age of 60 with a retirement portfolio of $100 million. However, his current portfolio stands at $50 million. So he needs to double his funds in the next 10 years to reach his retirement goal of $100 million. For that to happen, he needs to invest the current amount of $50 million in an investment vehicle that can offer him a rate of return of XX%, which can be calculated by using the Rule of 72.

*Rate of return required= 72/number of years*

= 72/10

= 7.2

Thus, Peter needs to invest in an investment vehicle that can offer him a rate of return of roughly 7.2% for achieving his retirement goal of $100 million.

## **How does the Rule of 72 work?**

Let’s go deeper to see how the Rule of 72 actually works.

The rule is based on the formula of compound interest. The compound interest formula is:

*Final amount= Initial investment * (1+r) ^n*

Where:

r= rate of return

n= number of years

Thus, if we want to double the investment, then we can input the numbers as below:

2= 1 *(1+r) ^n

To solve this, we need to use logarithms. A logarithm is the exponent or power to which a base must be raised to yield a given number. Mathematically, x is the logarithm of y to the base b if b^x = y, in which case we can write x = log_{b} y.

Similarly, taking natural logs on both sides of our equation, we get

log 2= log[(1+r) ^n]

log 2= n* log(1+r)

0.693= n*log(1+r)

Approximating, log(1+r) = r, we can simplify this further,

log 2= n*r

Thus, we get n*r= 0.693

or n = 69.3/r

This is called the **rule of 69.3**.

However, even though dividing 69.3 by the rate of return gives us an approximation of the number of years required, 69.3 is not an easy number to divide. The number 72 offers the benefit of having a larger number of factors such as 2, 3, 4, 6, 8, and 9. Also, using 72 gives more precise results for interest rates between 6% to 10% – a common range of rates on several investments. For that reason, it is immensely easier and more popular to use 72 instead of 69.3 in the rule’s formula.

## **Who invented the Rule of 72?**

The magical property of the number 72 is worth appreciating. But who invented this rule? The earliest known reference points to Luca Pacioli in 1494. He referenced this rule in his comprehensive mathematics book called *Summa de Arithmetica*. However, there is no derivation of the rule in his book, which makes some believe that the rule pre-dates Pacioli’s book.

## **How to use the Rule of 72?**

Apart from calculating the time it will take to double your investment, the Rule of 72 can be used in other decisions and places as well.

**Rule of 72 to calculate inflation effects**

It is a well-known fact that inflation is a silent wealth destroyer that reduces the purchasing power of money over time. The Rule of 72 can also help you calculate the effect of inflation on your investments. The Rule of 72 can be used for calculating how much time it takes for a portfolio to halve in purchasing power value due to inflation. Let’s see this with a practical example.

Tony, age 30, has money in a bank that offers him a near-zero rate of return on his deposit. He wants to know what effect a 4% inflation rate will have on his deposit, and how many years it will take for the purchasing power of his deposit to be reduced to half. This can be done by using simple calculations. All we have to do is divide the number 72 by the inflation rate.

Number of years = 72/4 = 18 years

Thus, by the time Tony reaches the age of 48 (calculated as 30+18), his deposit will be able to help him buy only half as much as he can do with it today. That means if Tony can buy daily supplies for two years with the amount he has today, in 18 years the same amount will help him purchase supplies only for a year.

**Calculating the effect of fees on investment gains**

Another interesting area where the rule can be used is in calculating the effect of fees on investment gains. A mutual fund that charges 2% annual expense fees will reduce the investment principal to half in 36 years. This is calculated as,

Time in years (n) = 72/r = 72/2 = 36 years

Similarly, if we increase the expense fees to 3%, the time taken for the same effect will be 24 years.

**Understand your debt **

Just as compounding works for your investments, it works against you when you have debt.

As an example, if you have a credit card debt with an annual interest rate of 20%, you can use the rule of 72 to calculate the time in which the amount you owe will be doubled:

n = 72/r = 72/20 = 3.6

So, the amount you owe will double in just roughly three and a half years!

**Variations of the rule for more precise results**

Rule of 72 only works on compounded rate of return and it is most accurate in the rate of return range of 6% to 10%. As the rate of return diverges, the results obtained from the rule also deviate from the actual calculation. For rates outside the range of 6% to 10%, one can simply add or subtract 1 from 72 for every 3 points that the rate diverges from 8%. For example, the rate of 14% annual compounding interest is 6 points higher than 8%. So you should add 2 (calculated as 6 divided by 3) to 72, which leads to using the Rule of 74 for higher precision. Similarly, for a 5% rate of return, we have to reduce 1 from 72, which leads to the Rule of 71.

## **Rule of 72 Calculator Excel**

Let us now calculate the number of years taken for an investment to double with the help of the Rule of 72. You can cross-check how accurate the results are from the actual number by using a more advanced calculator.

Rate of return (%) | Number of years to double investment using rule of 72 | Number of years to double investment |

1 | 70.000 | 69.661 |

2 | 35.000 | 35.003 |

3 | 23.667 | 23.450 |

4 | 17.750 | 17.673 |

5 | 14.200 | 14.207 |

6 | 12.000 | 11.896 |

7 | 10.286 | 10.245 |

8 | 9.000 | 9.006 |

9 | 8.000 | 8.043 |

10 | 7.200 | 7.273 |

11 | 6.636 | 6.642 |

12 | 6.083 | 6.116 |

13 | 5.615 | 5.671 |

14 | 5.286 | 5.290 |

As the table above shows, the Rule of 72 gives fairly accurate results. The green highlighted cells are the rates (6% to 10%) where the rule works best. For rates above and below this range, 1 has been added to or subtracted from 72, as discussed above. For example, for a rate of return of 14%, we need to use 74 in the formula to get more precise results.

For the Excel calculation, you can use the following:

log 2= n* log(1+r)

So, n = log 2/log(1+r)

Formula in excel: LN(2)/LN(1+r)

Thus, the Rule of 72 is quite handy to figure out how much time it roughly takes to double your investment.

## **401(k) and the Rule of 72**

The Rule of 72 can be very handy while doing retirement planning as it can tell you how many times your investment can double over the long run. To best understand this, let us see an example.

Frank, age 30, has an initial retirement fund of $100,000 today. He intends to invest it in a product that is offering him a rate of return of 9%. He plans to retire at the age of 62. Let us see how many times his initial investment of $100,000 will double until he reaches that age.

By using the Rule of 72, the number of years it will take for the investment to double with a rate of return of 9% comes out at 8 years (calculated as 72 divided by 9). So in 8 years, Frank’s initial investment will double to $200,000. With a rate of return of 9% for another 8 years, this amount of $200,000 will double to $400,000. By the time Frank retires at the age of 62, his initial investment of $100,000 would have doubled 4 times to reach $1.6 million. Thus, Frank will be a millionaire by his retirement with just an initial investment of $100,000. Kudos to Frank! This is a practical use of the Rule of 72 in retirement planning.

Notably, the Rule of 72 should not be confused with Rule 72(t) of the IRS, which outlines the process of making early withdrawals from a 401(k) account without paying extra penalties.