How to Double your Money – The Rule of 72

Ever wonder how long it would take to double your money? At first glance, many people might think, “Well, I need a 100% increase, so I’ll just divide 100 by my annual interest rate.” Try this method with 10% (just as an example) and you’ll get 10 years. This is wrong, because it doesn’t consider the power of compounding interest. The real length of time is actually quite a bit less.

The Rule of 72

Enter the rule of 72. Simply divide 72 by your expected average annual interest rate! This gives you a very good estimate of how long it will take to double your money, including the effect of compounding interest. So, for our example above, to double your money given a 10% yearly return, it would take a little more than 7 years. If you could manage a 24% interest rate, it would only take 3 years to double your money! Below is a table summarizing this for several rates.

Rule of 72 Examples

The Rule of 72 Derived

I’ll explain it in words, as the equations are written below. First look at Equation A. This equation says I want a value (V2) to be equal to another value (V1) times itself plus an interest rate (1 + r). Because I want to compound this over multiple years, and it will always be multiplied by the resultant, this term is raised to the t power to express the number of times (or years) it is compounded.

Next, look at Equation B. This states the second value will be two times the first value (ie. we want our initial value to double).

Rule of 72 derived

If we substitute Equation B into Equation A, we get the next line. The V variables then cancel, and we only have to solve for t. This is done by taking the natural logarithm of both sides. The natural logarithm of 2 is approximately 0.693. The natural logarithm of any small number plus one can be approximated as that small number (in this case, r). Because our rates are in percentages, I multiplied numerator and denominator by 100 to get the 69.3.

Now we have an expression showing the time, t, to double our money is about 69.3 divided by our rate, r. Because 69.3 is not easy to use with mental math and approximations, 72 is chosen. We’re merely looking for an estimate, and 72 is a close number that’s easily divisible by many numbers (ie 2, 3, 4, 6, 8, 9, 12, 18, 24, 36). Thus, the rule of 72!

To Triple or Quadruple your Money

You can use the same method to figure out how to double or quadruple your money. For tripling, divide your rate into 110; for quadrupling, divide your rate into 140. How were these figured? Hint: ln(3) = 1.10, ln(4) = 1.39.

2 Comments

  1. David

    July 9, 2010
    2:07 PM

    Thanks for the explanation. The only quibble I have is that your reasoning of why 72 is used instead of 69.3 (or 69, or 70) is not too convincing. No doubt 72 is easier to use in mental arithmetic than 69.3, but it’s less obvious that it’s easier to use than 70. The real reason is that 72 is more accurate for the values of r that are likely to arise in actual calculations. For r=5%, ln(1+r) = .9758*r and for r=10%, ln(1+r)=.9531*r So for 5% interest 71.0 would be a much better number to use than 69.3, and for 10% interest 72.7 would be better than 69.3. So for a wide range of realistic interest rates, 72 leads to superior approximations than 69.3.

  2. la la

    October 24, 2011
    9:09 AM

    it is good to know about the rule of 72!!!!!!!!!

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