The Motley Fool Discussion Boards

Previous Page  
Investing/Strategies / Mechanical Investing 

URL:
http://boards.fool.com/howverysloppyofmecorrection1thecagrof26107599.aspx


Subject: Re: CAGR/GSD Limits of Blends  Date: 11/16/2007 6:51 PM  
Author: JuanBobsDad  Number: 203975 of 251824  
How very sloppy of me! Correction: 1. The CAGR of a blend cannot be larger than the realdollar arithmetic average of all the component returns. 2. The GSD of a blend can be smaller than any of the component GSDs; the blend GSD can approach zero. a. The reduced GSD goal of a blend can be achieved if, and only if, at least two of the components are negatively correlated (i.e., the components vary “out of phase” so that when one tends to go up, another tends to go down). Var(X) = sum of the series of xu values Var(Y) = sum of the series of yu values Cov(X,Y) = sum of the series of (xiu)*(yiu) values Cov(X,Y)=0 is the definition of X and Y being independent (i.e., uncorrelated). In this case the blend Variance is equal to the weighted sum of the component Variances. If yi – u =  (xi – u), Cov(X,Y) =  Var(X) =  Var(Y) and Var blend = 0 (the covariance term is doubled!). However it remains that the covariance term must be negative to reduce the blend’s variance from the weighted sum of the component variances. Warrl, when you say “It is sufficient that the correlation be less than 1.”, I think you have the correlation coefficient in mind. The correlation coefficient is the Cov(X,Y) divided by the product of the standard deviations, stddev(X)*stddev(Y). Again, the correlation coefficient must be negative to achieve the goal of a blend (diversification). JimZipCode, You say, “In theory, a blend could exceed the CAGR of all its constituents.” You are indeed correct. The simple illustration is two screens with GSD but low CAGRs that have a correlation coefficient = 1. The blend of these two will have zero variance which also means that the blend CAGR is equal to the arithmetic average of the realdollar returns – a value that indeed can be larger than either of the constituent CAGR values. 

Copyright 19962014 trademark and the "Fool" logo is a trademark of The Motley Fool, Inc. Contact Us 