Compound Interest - Rule of 70

Date: Tue, 29 Nov 1994 14:12:30 -0800 (PST)
From: Mizuki Nishisaka
Subject: Rule of 70 

There is a famous "Rule of 70" for compounded interest.  
What is it and why does it work?

Mizuki Nishisaka

Date: 7 Dec 1994 17:44:55 GMT
From: Dr. Math
Subject: Re: Rule of 70

Hey Mizuki,

The question is:

Given a compounding rate 1.r ( r is some two-digit number like
08 15 12 04  etc.),  how many years will it take to double if it is
compounded yearly?

For example, if your savings account has an 8% interest rate, then
your compounding rate is 1.08.  The way to do this is the following:

Let p0 be your initial investment and p1,p2,p3,... be how much
you have after each year.  So  p1 = (1.08)p0
                      p2 = (1.08)(1.08)p0
                      p3 = (1.08)(1.08)(1.08)p0
                      p4 = (1.08)(1.08)(1.08)(1.08)p0
                      p5 = (1.08)(1.08)(1.08)(1.08)(1.08)p0

So pN equals p0 times (1.08)^n  - so the question we are asking is when does 

         (1.08)^n = 2                     Using natural logs we can find that 

                     n = ln(2)/ln(1.08)

and it turns out that this is very close to 70/r.

So 70/r can be used to approximate the number of years needed to
double your money if r is your interest rate. 

Here are a few comments on why this is true.

Precisely what we are saying is that  ln(2)/ln(1 + r/100) is approx. 70/r.

Or,  the value of the first derivative of the lefthand side is approximately
seventy.  This is true; however, for calculation purposes it is probably
easier to show that the derivative of the reciprocal  { ln(1 + r/100)/ln(2) } 
is 1/70.  Once you know this  it is enough to show that these two equations 
are close at one point, and you can say that they will be reasonably close 
at many points, near r = 10 for instance.

     Hope that this helps you. Thanks and bye.

     Ethan, Doctor ON CAll 

Date: 28 Sep 2000 17:19:55 GMT
From: Dr. TWE
Subject: Re: Rule of 70

Actually, in financial circles, it is usually referred to as the "rule 
of 72." Dividing the value 72 by the interest rate gives closer 
approximations when dealing with rates around 6% to 8%, which are 
typical in the financial industry. You can find out more about the rule 
of 72 by typing "rule of 72" without the quotation marks in our Ask Dr. 
Math searcher at:

   http://mathforum.org/mathgrepform.html   

When dealing with exponential growth in biology, the rule of 70 is more 
often used. When the growth rate is 3% or less, then dividing the value 
70 by the growth rate gives closer approximations than 72.

I hope this helps!

-Doctor TWE
 http://mathforum.org/dr.math