Principles of Pollex: The Rule of 72/Rule of 78’s

(In case you are confused by the headline: a principle is a rule, and pollex is an obscure term for thumb. Therefore, we’re talking about Rules of Thumb.)

In this installment of our ongoing Principles of Pollex series, we’re going to talk about two Rules from the financial world that are actually real, true, undisputed Rules, rather than the guidelines with dubious proof that we’ve talked about before.  These two Rules are not open to interpretation.  The first is about investments, and the second is about loans.  Both are useful in their own ways…

Rule of 72

The Rule of 72 is a quick and easy way to determine when an invested amount will double in value, given a particular fixed rate of return.  Please take note that this only works with a fixed rate of return.  The actual formula is as follows:

72/R = Y (where the divisor R = the fixed rate of return and result Y = the number of years to double the value of an invested amount)

So, if you were to invest \$1,000 at a rate of 4%, 72 divided by 4 equals 18. Therefore it would take 18 years to double in value to \$2,000 at the fixed rate of 4%.

Another way to use this formula is to determine the fixed rate of return that you would need to achieve in order to double the value of an investment within a particular known timeframe.  This is possible because the formula can be rewritten as 72/Y = R as an equivalent.  Here’s an example:

If you had \$1,000 and you wanted to double the value to \$2,000 within 10 years, you divide 72 by 10 years, and the result is 7.2. So, you would have to achieve at least a 7.2% return to accomplish a doubling of your investment in 10 years.

Rule of 78’s

This rule is useful for calculating loan interest being paid with each payment of a loan, or the accumulated amount of interest paid to date. This applies primarily to mortgage loans or other loans that are not based on simple interest calculations, like credit card debt. The name of this rule comes from the fact that when the numbers 1 through 12 are added together, the result is 78.   But why is that important?  Don’t fret – we’re getting to that part!

You’ve heard that most of the interest is paid first in a loan, right?  It’s true: interest in common Rule of 78’s loans (also called “sum of the digits” loans) is loaded toward the front of the pay-back cycle.  The way that interest is paid off in a 1-year (12-month) loan is as follows:  in the first month, 12/78ths of the interest is paid; in the second month, 11/78ths; third month, 10/78ths; and so on until 1/78th is paid in the final month.  The remainder of each fixed amount of payment each month goes toward the principle.

So using the Rule of 78’s we can figure out how much interest has been paid at any one time (assuming the payments are paid exactly as prescribed, no additional payments or late payments have been made) by adding up the Rule of 78’s factors up to the present month.  If we know that the total finance charge for our one-year loan is \$200, and we’ve made four payments, we can see that we’ve paid \$107.69 in interest so far.  This is calculated as:

(12+11+10+9) / 78 * \$200 = \$107.69

But what if our loan is for 36 months instead of just one year?  This is where the alternative name, “sum of the digits” comes into play… Of course adding up the months of payment won’t equal 78 – when we add 1 through 36 together we get 666 (ominous, I know!).  Following what we discovered about a 12-month loan, we know that in the first month, 36/666ths of the total interest will be paid; during the second month, 35/666ths; and so on.  Knowing what our denominator is now, we can cipher the amount of interest that will be paid with the 20th payment – 17/666ths – for example.

Keep in mind that the Rule of 78’s calculations are only useful in “pre-computed” loans – such as auto loans or mortgages.  For revolving loans (like a credit card), you pay interest currently each month (or the interest is added currently if you’re not paying the interest amount).

Have fun and “rule” your financial universe!