Magic Numbers: Reduce the Math of Annuities to Simple Arithmetic
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In all these situations, the stream of money being withdrawn from savings is an annuity.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.
In this article
- Present Value of Level Annuities
- Future Amount of Level Annuities
- Present Value of Geometrically Increasing Annuities
- Future Value of Geometrically Increasing Annuities
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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.
These rules may also be used to determine the approximate number of years it will take to deplete savings based on the present value of a fund (e.g., retirement savings) or to determine the number of years it will take to accumulate a future amount (an accumulation), given specified annual withdrawals or savings, respectively. Further, given a specified numbers of years, you can determine what the approximate annual withdrawal or savings should be. All of these numbers can be determined using a simple hand-held calculator. If a fractional multiple is suggested, the average of the magic numbers of the two surrounding multiples may be used. Further, the magic numbers in Table 1 were derived using an 8% interest rate, or rate of return. For rates other than 8%, there is a 1% error for every 2% difference between the interest rate and 8%. However, the error in the magic numbers is less than 3.5% for interest rates between 2% and 15% and would have a negligible effect on approximations.
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