Magic Numbers: Reduce the Math of Annuities to Simple Arithmetic

by Robert Muksian

Magic Numbers: Reduce The Math Of Annuities To Simple Arithmetic Splash image

Individuals planning for or approaching retirement commonly have questions about how long their savings will last, how much money they will be able to withdraw or how long it will take to save a certain amount.

In all these situations, the stream of money being withdrawn from savings is an annuity. In any final analysis, exact annuity formulas must be used to determine exact values. The appropriate formulas for answering questions about “how long” and “how much” require the use of logarithms. However, in a preliminary analysis or for long-range planning, reasonable approximations may be sufficient, because the future is not a certainty. The “magic numbers” 72, 114 and 167 may be used to obtain these approximations with simple arithmetic.

You may already be familiar with the Rule of 72, where the approximate number of years it takes for money to double is given by dividing 72 by the interest rate. Derived in the same manner are other, less-known rules: 114 divided by the interest rate for tripling money and 167 divided by the interest rate for quintupling money. With these three rules, magic numbers for multiples up to 15 are readily determined; they are displayed in Table 1. The “mavericks” are multiples of 7, 11 and 13, which are prime numbers and must stand alone.

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Robert Muksian is a professor of mathematics at Bryant University in Smithfield, Rhode Island.


Walter from California posted over 2 years ago:

Excellent article.

Makhtar from New Jersey posted over 2 years ago:

No comment really, this is a great tool for the novice investor like myself.

Lawrence from Maryland posted over 2 years ago:

I'm need to study this more as it allows a 25% increase in withdrawals (at an interest rate of 3% for 30 years) from retirement savings than the conventional "take no more than 4%.
The use of an interest rate of very low interest rate of 3% should adjust for many of the fluctuations in the market but to be sure I'm also using as the savings the lowest balance it has been in 5 years.

Peter from Utah posted over 2 years ago:

I have spread sheets and a computer. Don't need these rules anymore. This for the land line set.

John c from Illinois posted over 2 years ago:

The other magic number is tax which does not appear to be mentioned in the article.

W from Pennsylvania posted over 2 years ago:

An HP calculator will do all that without having to use all the math required here!
Don Joran, CLU, ChFC

George from Oklahoma posted over 2 years ago:

Helpful article. The PDF link in the last paragraph doesn't seem to be working:

"For more detailed math on magic numbers and their use in annuities, a PDF file is available here ("

Would very much like to see the Magic Numbers Appendix when the link is fixed. Thanks!

Charles from Illinois posted over 2 years ago:

The link for the appendix has been fixed. -Charles Rotblut

Dan from North Carolina posted over 2 years ago:

A fine article for individuals investing in fixed income securities only. However, these projections are guaranteed to mislead investors savings, withdrawals, and longevity rates since variable securities, like stocks, funds, etc., will never produced a reliable fixed rate of return in perpetuity. Investors are better off finding a bootstrap Monte Carlo program to plan for investing a withdrawals rates.

Allen from Pennsylvania posted about 1 year ago:

Too complicated for even the experienced investor and pre-retiree's. I would like to see a table..One Table that explains the following: If I get 4% return on my protfolio overall this year, how much can I withdraw at the end of 'this' year so that my money will last the 29 remaining years of the 30 years? I think a table would make it more simple and offer a guide for people to use. Adjust for inflation and taxes and I think you could have an article that is helpful to all and only one page! Please try again..Thank you..

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