Compound interest and doubling your money

There are some basic rules of finance whose understanding is integral to the proper management of money.  The first rule has been to understand the concept of the time value of moneyInterest rates drive all things finance; from credit card debt and house buying to savings accounts and bond investments.  The concept of the time value of money is rooted in mathematics and can be cumbersome to commit to memory.  The good news is that there has long been a rule that is easy to commit to memory and serves as the 95% solution to figuring out how much an asset can make you or how much a loan can cost you.

The rule is called “The divide by 72 rule“.  If you already know of this rule, you’re ahead of many.  However, do you know why that rule works, or even whether it actually does or not.  Anything based math should not be taken on faith, but instead proved through relationships.  The 72 rule states:

To quickly calculate the time in years to double your money in an investment due to compounding interest payments, divide 72 by the annual interest rate.  For example an 8% interest rate would take (72/8) 9 years to double the money invested.

This can work with interest rates that are compounded annually or even in shorter terms, so long as the annual effective rate is used in the calculation.

Rate Rule of 72 Annual Delta %
1% 72.0 69.7 3.2%
6% 12.0 11.9 0.9%
11% 6.5 6.6 -1.5%
16% 4.5 4.7 -3.8%
21% 3.4 3.6 -6.1%
26% 2.8 3.0 -8.3%
31% 2.3 2.6 -10.5%
36% 2.0 2.3 -12.7%

>>>The above table shows a comparison between the rule of 72 and annually compounded interest rates.  Notice that the error is low for interest rates that we commonly calculate.  The formula for the time to double your money is Time=ln(2)/ln(1+rate), where the time is in years and the interest rate is expressed as a percentage.  The divergence at higher interest rates from the rule of 72 is cause by the rule of 72 actually being tied to the formula for continuously compounding interest.  This is where interest is always compounding upon itself.  The formula for doubling you money in this case in Time=ln(2)/rate.  This formula tracks perfectly with a rule of 69.3 as shown below.  The reason that 72 is used is because “head math” with 72 works easier as 72 has plenty of multipliers (1,2,3,4,6,8,12…and so on) that are common interest rates.

Rate Rule of 69.3 Annual Delta %
1% 69.3 69.3 0.0%
6% 11.6 11.6 0.0%
11% 6.3 6.3 0.0%
16% 4.3 4.3 0.0%
21% 3.3 3.3 0.0%
26% 2.7 2.7 0.0%
31% 2.2 2.2 0.0%




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