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Investing/Strategies / Mechanical Investing
|Subject: Re: CAGR/GSD Limits of Blends||Date: 11/16/2007 6:51 PM|
|Author: JuanBobsDad||Number: 203975 of 256609|
How very sloppy of me! Correction:
1. The CAGR of a blend cannot be larger than the real-dollar 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 x-u values
Var(Y) = sum of the series of y-u values
Cov(X,Y) = sum of the series of (xi-u)*(yi-u) 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.
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).
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 real-dollar returns – a value that indeed can be larger than either of the constituent CAGR values.
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