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Seventy-two and 115: What Do Logs Have to Do with Doubling and Tripling Your Money?

Date: 02/19/2010 at 01:52:09
From: GEmma
Subject: Rule of 115

How does the rule of 115 relate to the rule of 72?

I am doing an assignment, and have to justify my work.

I know that I have to use logs to find time, however I am just not 
understanding why it's 115.



Date: 02/19/2010 at 11:51:54
From: Doctor Carter
Subject: Re: Rule of 115

Hi Gemma -

Thanks for writing to Dr. Math.  This is a very interesting 
question!  

As you know, the rule of 72 says that if you have an investment 
earning an annual rate of return of I percent, the number of years it 
will take your investment to double in value is *approximately*
72/I.  Example:  If you invest at 10%, it will take your investment 
approximately 7.2 years to double in value.

Similarly, the rule of 115 says that if you have an investment 
earning I percent every year, the number of years it will take your 
investment to triple in value is *approximately* 115/I.  Example:  If 
you invest at 10%, it will take your investment approximately 11.5 
years to triple in value.

Where do the numbers 72 and 115 come from?  To answer this, we need 
to look carefully at the compound interest formula.  

Let's write i for I/100, the annual interest expressed as a decimal.  
The compound interest formula (for annual compounding) tells us that 
if the original amount invested is P, the value after N years is P 
times (1 + i)^N.  The number of years it takes money to double is 
therefore the value of N for which

   (1 + i)^N = 2

Taking logarithms, we see that the number of years it takes money to 
double is 

   N = log(2)/log(1 + i)

This formula is true no matter what base is used for the logarithms.  
But for what I want to do next, it is important that log() be 
interpreted as the natural logarithm (log to the base e), because 
for the natural logarithm, we have a formula (valid for i less than 
1) which says

   log(1 + i) = i - i^2/2 + i^3/3 - i^4/4 + ...

This formula is called a Taylor series, and is derived using 
calculus.  If you know about these things, great!!  If not, you'll 
have to take my word.  

We can factor i out of the formula for log(1 + i):

   log(1 + i) = i (1 - i/2 + i^2/3 - i^3/4 + ... )

If i is between 0.0 and 0.2, we have the following APPROXIMATION:

   log(1 + i) = i (1 - 0.1/2 + 0.01/3 - 0.001/4 + ... ) = i (0.9531)

The number of years it takes money to double is therefore 
APPROXIMATELY

   N = log(2) / ((0.9531) i) 

Now,

    log(2)/(0.9531) = 0.7273

This says that the doubling time is approximately 0.7273/i, which is 
72.73/(100 i). Remembering that the interest I in percent is I = 100 
i, we see that the doubling time is approximately 72/I.  That's the 
rule of 72!

To obtain the rule of 115, we can go through a very similar 
argument.  I'm not going to repeat all the details, but if you work 
through it yourself you will learn that the number of years it takes 
an investment to triple in value is approximately

   N = log(3) / ((0.9531) i)

And

   log(3)/0.9531 = 1.1527

So the investment triples in value in approximately 1.1527/i years.  
Since the interest rate I in percent is 100i, this says the tripling 
time is approximately 115/I years.

That's it!  That's where those numbers come from!

- Doctor Carter, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Interest
High School Logs

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