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November 18th, 2007, 04:47 PM  #1 
Member Joined: Aug 2007 Posts: 93 Thanks: 0  A Basic Problem
For each period n an investment will either gain or lose 10% with probability of 50% each way. As n goes to infinity, what is the geometric mean return on the investment?

November 18th, 2007, 04:58 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,640 Thanks: 2082 
The mean return doesn't depend on n; it's zero.

November 18th, 2007, 05:26 PM  #3  
Member Joined: Aug 2007 Posts: 93 Thanks: 0  Quote:
The mean return on the investment is not zero, as is proven by if n=2 the mean result is (1+10%)(110%) = 1%  
November 18th, 2007, 05:44 PM  #4 
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 
n = 0: value 1, mean = 1 n = 1: value 0.9 (1/2), 1.1 (1/2), mean = 1 n = 2: value 0.81 (1/4), 0.99 (1/2), 1.21 (1/4), mean = 1 An inductive proof should show that the mean remains at 1. I believe the median tends toward 0, but I'd have to check. 
November 18th, 2007, 05:56 PM  #5  
Member Joined: Aug 2007 Posts: 93 Thanks: 0  Quote:
 
November 18th, 2007, 09:48 PM  #6  
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  Quote:
Your 'law of large numbers' shows the median for even n, not the mean.  
November 19th, 2007, 06:30 AM  #7  
Member Joined: Aug 2007 Posts: 93 Thanks: 0  Quote:
If this binomial distribution is taken to its limit, can anyone formulate the difference between the expected geometric mean return of the investor (i.e. the compounded rate of return) and the expected arithmetic average return (which is zero)? What continuous distribution describes the wealth of the investor as n goes to infinity?  
November 19th, 2007, 11:11 AM  #8  
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  Quote:
Quote:
The arithmetic mean is 1. The geometric mean is 0. The wealth distribution is uniformly 0.  
November 19th, 2007, 05:57 PM  #9 
Member Joined: Aug 2007 Posts: 93 Thanks: 0 
Actually the geometric mean return diverges from the arithmetic mean by half the variance the resulting wealth distribution is lognormal If you think about it this is a potent argument for minimizing your volatility by diversifying your investments  it actually will lead to a slightly higher compound return. 
November 20th, 2007, 05:53 AM  #10  
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  Quote:
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