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# The rule of 72, or how to impress your friends (and your banker)

A sum of money is invested at compound interest when the interest received for each period is added to the the initial sum (often referred to as the initial capital) to increase the amount of money which earns the interest. In other words, the calculation of compound interest takes into account that the interest you are earning will be added to the savings you have initially invested and earn interest itself. According to Warren Buffet, the famous investor also known as the “Oracle of Omaha” and one of the wealthiest men in the world, compound interest constitute the key to the creation of his fortune and a relevant and very beneficial force for anyone who wishes to put his / her money to work and see their savings grow. But how can one illustrate its effects in a simple and tangible way, so that we can all master this financially super powerful concept? Behold the rule of 72!

## Definition and practical examples of the rule of 72

The rule of 72 was discovered during the Renaissance and is described in Summa de arithmetica, geometria, de proportioni et de proportionalita by Luca Pacioli, published in 1494. Einstein developed a theory on it and stated that ”Compound interest is the greatest force in the entire universe.”

It can be summarised as follows : if a capital amount is placed at a given interest rate of t % per period (typically on an annual basis), it will require 72/t periods to double it.

Let’s provide a few practical savings examples. Imagine that I invest my savings in a French Livret A savings account, earning 0.75% per year. According to the rule of 72, it will take me 72 / 0.75, or 96 years to double my money.

Now imagine that as a result of a more dynamic allocation of my savings, my money is earning 4% per year. Again using the rule, I now only need 72 / 4, or 18 years to achieve the same result!

What interest rate or return do I need to double my money in 6 years? Very simple : according to the rule of 72, I will need to find an investment yielding 12% per year, because 72/12 = 6.

The mathematical theory behind this rule can be found below1. In this explanation we also explain why, in the current very low interest rate environment, it is mathematically better to apply the rule of 70 instead of 72, but the principles remain the same. You will easily understand that this rule enables you to impress your audience. Imagine the look on your friends’ faces when you state that you can calculate the number of years that are required to double any sum of money, invested at any given rate!

But more importantly, it allows you to demonstrate how even small differences in yield can have a considerable impact on your gains over time.

## How hard are your savings working for you?

The rule of 72 (or rather 70, given the low interest rate environment) allows anyone to swiftly determine and with accuracy how many years it will take to double a certain sum of money, according to the interest rate at which it is invested. In the current low interest rate environment, it sheds light on the fact that for certain very widely used savings products, such as the Livret A (yielding 0.75%) or guaranteed life insurance contracts (offering a yield of 1.6% on average) the time required to double the initial capital exceeds 90 and 40 years respectively. That does not necessarily mean that these are bad investments, but - in Cashbee’s opinion - it does make it clear that it is worthwhile to consider more rewarding alternative investments for part of one’s savings, particularly if one is putting this money aside for the longer term.

We have not found a better formulation than that offered by Wikipedia to explain the rule of 72 mathematically. We therefore provide you with it here : The rule of compound interest states that if an amount is invested during n periods at a rate of t, it is multiplied by (1+t)[n]. To double that amount, it follows that (1+t)n = 2. By applying the natural logarithm of each member, it follows that n*ln(1+t) = ln(2), or 0.693… And if t is very small then ln(1+t) ≈ t. Thus, n ≈ 0.693 / t, and if t is expressed in percentages then n ≈ 69.3 / t. A rounded value of 70 should therefore be used when the interest rate is low, between 0.5% and 3%. The value of 72, which has given its name to the rule, is a more exact reference point for interest rates between 5% and 10%. As this number 72 has more divisors, the the mental calculation is simpler in this instance.

1. The mathematical explanation of the rule of 72: