**Casio fx-CG 50: Double the Investment, Rule of 72, and Rule of 69**

**Introduction**

Are you familiar with the Rule of 72? In finance, the Rule of 72 refers to a simple formula to determine approximately how many years it takes an investment to double in value.

The Rule of 72 is an approximate formula, stated as:

n = 72 ÷ I

We can determine the actual time value of money formula (without periodic payment):

FV = PV × (1 + I ÷ 100)^n

where:

FV = future value

PV = present value

I = interest rate

n = number of years

Let PV = 1 and at double the investment, FV = 2.

2 = (1 + I ÷ 100)^n

Solving for n:

ln 2 = ln [ (1 + I ÷ 100)^n ]

ln 2 = n × ln [ (1 + I ÷ 100) ]

n = ln 2 ÷ ln [ (1 + I ÷ 100) ]

This is an exact calculation.

There is another approximation, called the Rule of 69, and the approximation is this:

n = 69 ÷ I + 0.35

In Summary:

Exact Calculation: n = ln 2 ÷ ln [ (1 + I ÷ 100) ]

Rule of 72: n = 72 ÷ I

Rule of 69: n = 69 ÷ I + 0.35

**Comparison**

A spreadsheet, generated using a Casio fx-CG 50, is shown for rates from I = 2% to I = 20% in increments of 1%.

The Rule of 72 is the most accurate at I = 8%, with an error of 6.4 × 10^-3. At interest rate I ≈ 7.846871453, the Rule of 72 equals the exact calculation.

When I < 7.846871453, the Rule of 72 over-estimates. Other wise the Rule of 72 under-estimates.

For I = 2 to at least I = 20, the Rule of 69 under-estimates, but the difference shrinks as I increases.

Here is a graphical comparison for values of I = 5% to I = 15%.

**Source**

Thomsett, Michael C. __The Real Estate Investor's Pocket Calculator__ 2nd Ed. AMACOM: New York. 2018. ISBN 9780814438893 (paperback)

All original content copyright, © 2011-2021. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

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