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January 13th, 2015, 05:58 AM  #1 
Newbie Joined: Jan 2015 From: Australia Posts: 2 Thanks: 0  Mental Arithmetic
Hi All, I'm a new poster, hopefully this is in the correct section. I was reading a book about Warren Buffett, world famous investor. He was posed this question and answered it very quickly in his head "If the price of a painting goes from \$250 to \$50 million in one hundred years, what's the annual rate of return?" His answer was approx 13%, the interviewer was stunned how he knew it so quickly. The interviewer wanted to know how he did it. He said this "Buffett pointed out that any compound interest table would reveal the answer. Another way to approach the problem, said Buffett, was "to go by the number of times it doubles ( \$250 doubles about 17.6 times to get to \$50 million, a double every 5.7 years, or about 13 percent a year )." Simple, he seemed to say. So my question is what are the mental steps to be able to do this sort of calculation in your head? Thanks Everyone for your help. This is from the book by Robert Hagstrom "The Warren Buffett Portfolio" Last edited by greg1313; January 13th, 2015 at 06:05 AM. 
January 13th, 2015, 07:08 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,403 Thanks: 2477 Math Focus: Mainly analysis and algebra 
Probably he just knows certain values  he has a table memorised if you like. This is normal for someone who works in the field. When I worked on billing systems for telephone companies, I knew every area code in the UK and about half of the country dialling codes. 
January 13th, 2015, 08:18 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 19,542 Thanks: 1752 
This calculation is particularly easy if some base 10 logarithms are known by heart. I suspect that's how he did it, rather than the way he suggested.

January 13th, 2015, 10:46 AM  #4  
Senior Member Joined: Aug 2012 Posts: 2,010 Thanks: 574  Quote:
The Rule of 72 says that the doubling time at x% is 72/x. In other words at an interest rate of 10% compounded, a sum will double in 7 years. Now \$250 to \$50M is a factor of 200,000. The base2 log of 200,000 is about 17.6. This accords with the OP's narrative. Now if it doubled 17.6 times in 100 years, it doubled every 5 and a half years or so; 72/5.5 = 13+. That's exactly how finance types do these calculations. Rule of 72  Wikipedia, the free encyclopedia Last edited by Maschke; January 13th, 2015 at 11:19 AM.  
January 13th, 2015, 11:59 AM  #5 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
Rule of 72 was what I thought of, too.

January 13th, 2015, 01:37 PM  #6 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
I once had a student in a basic algebra course who had been a salesman for many years. He struggled with basic arithmetic problems unless they were phrased in terms of percentages. Then he could do them easily.

January 13th, 2015, 02:40 PM  #7 
Newbie Joined: Jan 2015 From: Australia Posts: 2 Thanks: 0 
Thanks everyone for the replies. Can someone please explain what this log base 2 is. Thanks 
October 19th, 2017, 06:39 PM  #8 
Newbie Joined: May 2017 From: Overland Park Posts: 3 Thanks: 0  Rule of 72
Warren was able to do this using the rule of 72 or the principle of 72 which helps a person calculate how long it takes for a sum of money to double at a given investment rate. This same principle works no matter what the time frame or multiplier. So if the money is going from 250 to 50 million, that's 200,000 times. Now, I'm not sure how Warren was able to do the log base 2 in his head...but he must have a method to be able to do this. I just prefer to use a 72 rule calculator to determine the years to double my investment. 

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