When It Comes to Doubling Your Money, the Rule of 72 is Handy

In an earlier posting Pennies a Day, we demonstrated how the power of compounding can be harnessed to build wealth. You may recall that when a penny is doubled every day for a month, it grows into more than \$10 million dollars! If you haven’t seen it, I encourage you to go back and read the article. Meanwhile, in the real world…

How do we calculate the time for an investment to double based upon a given annual interest rate?

The formula for determining this requires either a scientific or financial calculator or, using logarithmic or financial functions in a spreadsheet program.

However, there is another option that only requires simple division: The Rule of 72. While it is an approximation, it is uncannily accurate. It’s a quick, easy way to estimate the time required to double an investment on ‘the back of an envelope’ or ‘in your head’.

Here are some questions that can be quickly answered with the Rule of 72:

• If I have a CD with 2% interest, how long would it take to double my money?
• If I have a mutual fund that historically returns 6%, how long should I expect for my investment to double?
• If average annual inflation is 3%, how long will it take for prices to double – and for the dollar to only be worth half of what it is now?

To get an estimate divide 72 by the interest rate (in digits). For the questions above, we get:

• 36 years for the CD to double at 2% (72 ÷ 2)
• 12 years for the mutual fund to double at 6% (72 ÷ 6)
• 24 years for inflation to double at 3% per year (72 ÷ 3)

One caveat: The doubling factor for the CD and mutual fund does not take into account the impact of taxes or inflation.

Here’s the actual formula (or formulas) to calculate the exact period time for an investment to double:

Time to Double Investment = log(2) ÷ log(1 + (Interest Rate ÷ 100)) —Or— Time to Double Investment = ln(2) ÷  ln (1 + (Interest Rate ÷ 100))

‘log’ is the Base 10 logarithm, while ‘ln’ is the ‘natural logarithm’ based on the mathematical constant ‘e’

No need to worry if these terms sound like geeky gibberish.  These equations are mathematical tools that assist us in calculating compound interest. Note answers are the same whether you use the ‘log’ or ‘ln’ function.

Let’s go back to the 3 questions asked earlier. The actual answers do vary from the Rule of 72 approximations, but not by much:

• For the CD to double at 2%: log(2) ÷ log(1 + (2 ÷ 100)) = log(2) ÷ log(1.02) = 0.3010 ÷ .0086 = 35.00 years (vs. 36 years from the Rule of 72)

In a similar manner:

• 11.9 years for the mutual fund to double at 6% (vs. 12 years from Rule of 72)
• 23.45 years for inflation to double at 3% per year (vs. 24 years from Rule of 72)

Refer to the table below for an actual vs. Rule of 72 comparison for a range of interest rates. How about the flip question…

For my investment to double in a given number of years, what rate of return percentage would I need?

Using the Rule of 72, you would simply divide 72 by the number of years. Example: if you want an investment to double in 18 years: 72 ÷ 18 = 4% annual return. But the actual formula is a bit more involved:

100 x (10log(2) ÷ 18 – 1)     or   100 x (eln(2) ÷ 18 – 1)

Both formulas provide the same answer: 3.93%, quite close to 4% that the Rule of 72 gives.

Which approach would you prefer for a quick estimate? The Rule of 72 obviously wins hands down.

Hopefully you will find the Rule of 72 helpful in your financial planning and decisions. The Rule of 72 also reinforces the power of compounding that can and should be harnessed in your investments.

Regards, Paul

PS Here is a reference that explains the mathematical derivation for the Rule of 72:  https://betterexplained.com/articles/the-rule-of-72

 Interest Rate % Rule  of 72 Actual Calculated % Difference 1.00 72.00 69.66 3.36% 2.00 36.00 35.00 2.85% 3.00 24.00 23.45 2.35% 4.00 18.00 17.67 1.85% 5.00 14.40 14.21 1.36% 6.00 12.00 11.90 0.88% 7.00 10.29 10.24 0.40% 8.00 9.00 9.01 -0.07% 9.00 8.00 8.04 -0.54% 10.00 7.20 7.27 -1.00% 11.00 6.55 6.64 -1.45% 12.00 6.00 6.12 -1.90% 13.00 5.54 5.67 -2.34% 14.00 5.14 5.29 -2.78% 15.00 4.80 4.96 -3.22% 16.00 4.50 4.67 -3.64% 17.00 4.24 4.41 -4.07% 18.00 4.00 4.19 -4.49% 19.00 3.79 3.98 -4.90% 20.00 3.60 3.80 -5.31% 21.00 3.43 3.64 -5.71% 22.00 3.27 3.49 -6.11% 23.00 3.13 3.35 -6.51% 24.00 3.00 3.22 -6.90% 25.00 2.88 3.11 -7.28%

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