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)
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