## Sunday, November 21, 2021

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

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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